Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Sanjiva Dayal IIT Kanpur

Chapter 5

Applications Of Derivatives - all with Video Answers

Educators

Chapter Questions

Problem 1

Find the mean rate of change of the following functions in the interval $[1,2]$ :-
i. $\quad f(x)=x^{2}$. \{Ans. 3\}
ii. $\quad f(x)=x^{3}$. \{Ans. 7$\}$
iii. $f(x)=\sqrt{x}$. \{Ans. 0.414\}
iv. $\quad f(x)=\frac{1}{x} .$ \{ns. $\left.-0.5\right\}$
v. $\quad f(x)=e^{x}$. \{Ans. 4.67\}
vi. $f(x)=\ln x .\{$ Ans. $0.693\}$
vii. $f(x)=\sin x .\{$ Ans. $0.068\}$
viii. $f(x)=\cos x$. \{ns. $-0.956\}$

Nidhi Singhi

Nidhi Singhi

Numerade Educator

Problem 2

For what value of $a$, the mean rate of change of the function $f(x)=x^{2}$ in the interval $[a, a+1]$ is 2 ?

Nidhi Singhi

Numerade Educator

Problem 3

Find the instantaneous rate of change of the following functions at $x=1$ and also find Stationary points:-
i. $\quad f(x)=x^{2}$
ii. $\quad f(x)=x^{3}$
iii. $f(x)=\sqrt{x}$
iv. $f(x)=\frac{1}{x}$.
v. $f(x)=e^{x}$.
vi. $f(x)=\ln x$.
vii. $f(x)=\sin x$
viii. $f(x)=\cos x$

Nidhi Singhi

Numerade Educator

Problem 4

For what value of $a$, the mean rate of change of the function $f(x)=x^{3}$ in the interval $[-1, a]$ is equal to the instantaneous rate of change at $a ?$

Nidhi Singhi

Numerade Educator

Problem 5

Find the approximate value of the following:-
i. $\cos 31^{\circ}$.
ii. $\log 10.21$
iii. $\sqrt[5]{33}$.
iv. $\cot 45^{\circ} 10^{\prime}$

Rukhmani Jain

Numerade Educator

Problem 6

Test the following functions for monotonicity and find stationary points:-
i. $f(x)=2 x^{3}-9 x^{2}+12 x+29$.
ii. $\quad y=x^{3}-3 x^{2}+6 x-17$.
iii. $f(x)=x^{3}-3 x$.
iv. $\quad y=x^{2}(x-3)^{2}$.
v. $\quad f(x)=x^{9}+3 x^{7}+64$
v. $\quad f(x)=x^{9}+3 x^{7}+64$
viii. $f(x)=\frac{x}{\ln x}$
ix. $\quad f(x)=x^{x}$.
x. $\quad f(x)=x^{\frac{1}{x}}$.
xi. $\quad f(x)=x e^{x}$.
xii. $f(x)=x e^{-x}$
xiii. $f(x)=e^{\frac{1}{x}}$.
xiv. $f(x)=x e^{\frac{1}{x}}$
xv. $\quad f(x)=\log _{x}(\ln x)$.
xvi. $\quad f(x)=\ln (1+x)-\frac{2 x}{2+x}$.
xvii. $f(x)=x+\sin x$.
xviii. $f(x)=\frac{4 \sin x-2 x-x \cos x}{2+\cos x}, \quad 0<x<2 \pi$.
xix. $\quad f(x)=\tan ^{-1} x-\frac{1}{2} \ln |x|$.
xx. $\quad y=x-\cot ^{-1} x-\ln \left(x+\sqrt{x^{2}+1}\right)$.
xxi. $\quad f(x)=2 x^{2}-\ln |x|$.

Gaurav Kalra

Numerade Educator

Problem 7

Show that the function $f(x)=\sin x, \quad x$ is a rationalno.
$=x, \quad x$ is an irrationalno. has positive derivative at $x=0$ but $f(x)$ is not increasing at $x=0$.

Nidhi Singhi

Numerade Educator

Problem 8

Show that the function $\begin{aligned} f(x) &=\frac{x}{2}+x^{2} \sin \frac{1}{x}, \quad x \neq 0 \\ &=0, \quad x=0 \end{aligned}$
is continuous and differentiable in any neighbourhood of $x=0$ and $f^{\prime}(0)$ is positive but $f(x)$ is not increasing at $x=0$.

Nidhi Singhi

Numerade Educator

Problem 9

For what values of $b$ the function $f(x)=\sin x-b x+c$ is decreasing in the interval $(-\infty, \infty) ?$

Nidhi Singhi

Numerade Educator

Problem 10

If $g(x)=f(x)+f(1-x)$ and $f^{\prime \prime}(x)<0 ; 0 \leq x \leq 1$, show that $g(x)$ increases in $0<x<\frac{1}{2}$ and decreases in $\frac{1}{2}<x<1$

Nidhi Singhi

Numerade Educator

Problem 11

Given $g(x)=f\left(x^{2}-x-10\right)+f\left(14+x-x^{2}\right)$ and $f^{\prime \prime}(x)>0$ for all real $x$, except at finite no. of real values of $x$ for which $f^{\prime \prime}(x)=0$. Discuss the monotonicity of the function $g(x)$. \{ns. $(\infty,-3)$ decreasing, $\left(-3, \frac{1}{2}\right)$ increasing, $\left(\frac{1}{2}, 4\right)$ decreasing, $(4, \infty)$ increasing $\}$

Nidhi Singhi

Numerade Educator

Problem 12

Investigate the following functions for extremum at $x=0:-$
i. $\quad f(x)=\sin x-x$.
ii. $\quad f(x)=\sin x-x+\frac{x^{3}}{3 !}$.
iii. $f(x)=\sin x-x+\frac{x^{3}}{3 !}-\frac{x^{4}}{4 !}$.
iv. $\quad f(x)=e^{\frac{1}{x}}, \quad x \neq 0$
$=0, \quad x=0 .$
v. $f(x)=\cosh x+\cos x$.
vi. $f(x)=\cos x-1+\frac{x^{2}}{2 !}-\frac{x^{3}}{3 !}$.
vii. $f(x)=\cos x-1+\frac{x^{2}}{2}$.
viii. $f(x)=x+x^{\frac{2}{3}}$.
ix. $f(x)=x^{2}+x^{\frac{1}{3}}$.
x. $f(x)=x^{\frac{4}{3}}+2 .$
xi. $f(x)=x^{\frac{5}{3}}-3 .$

Gaurav Kalra

Numerade Educator

Problem 13

Test the following functions for extremum:-
i. $f(x)=2 x^{3}-15 x^{2}-84 x+8$.
ii. $\quad f(x)=x^{3}-6 x^{2}+9 x-8$.
iii. $\quad f(x)=-\frac{3}{4} x^{4}-8 x^{3}-\frac{45}{2} x^{2}+105$.
iv. $\quad f(x)=\frac{3}{4} x^{4}-x^{3}-9 x^{2}+7$
v. $\quad f(x)=x^{4}-8 x^{3}+22 x^{2}-24 x+12$.
vi. $\quad f(x)=x(x+1)^{3}(x-3)^{2}$
vii. $\quad f(x)=\frac{x^{2}-3 x+2}{x^{2}+2 x+1}$.
viii. $f(x)=3 \sqrt[3]{x^{2}}-x^{2}$.
ix. $f(x)=\sqrt[3]{(x-1)^{2}}+\sqrt[3]{(x+1)^{2}}$.
x. $f(x)=-2 x, \quad x<0$
$=3 x+5, \quad x \geq 0 .\{$
xi. $\begin{aligned} f(x) &=2 x^{2}+3, \quad x \neq 0 \\ &=4, \quad x=0 .\{\text { Ans. maxima at } 0\} \end{aligned}$
$f(x)=\frac{50}{3 x^{4}+8 x^{3}-18 x^{2}+60}$
$f(x)=\sqrt{e^{x^{2}}-1}$
$f(x)=x e^{x}$
$f(x)=x^{4} e^{-x^{2}}$
$f(x)=x^{2} e^{-x}$
xvii. $\quad f(x)=\frac{4 x}{x^{2}+4}$
xviii. $\quad f(x)=-x^{2} \sqrt[5]{(x-2)^{2}}$
xix. $\quad f(x)=\frac{14}{x^{4}-8 x^{2}+2}$
$f(x)=\sqrt[3]{2 x^{3}+3 x^{2}-36 x}$
xx. $\quad f(x)=\sqrt[3]{2 x^{3}+3 x^{2}-36 x}$.
xxi. $\quad f(x)=x^{2} \ln x$
xxii. $\quad f(x)=x \ln ^{2} x$
xxiv. $\quad f(x)=|x|+|x-1|+|x-2|$.
xxv. $\quad f(x)=\sin ^{4} x+\cos ^{4} x, 0<x<\frac{\pi}{2}$
xxvi. $\quad f(x)=\sin 2 x-x,-\frac{\pi}{2} \leq x \leq \frac{\pi}{2}$.
xxvii. $f(x)=\sin x+\frac{1}{2} \cos 2 x, 0 \leq x \leq \frac{\pi}{2}$.

Gaurav Kalra

Numerade Educator

Problem 14

The function $f(x)=a \sin x+\frac{1}{3} \sin 3 x$ has maximum value at $x=\frac{\pi}{3}$. Find the value of $a$.

Nidhi Singhi

Numerade Educator

Problem 15

Given $f(x)=|x-2|+\ln \left(a^{2}-1\right), \quad x<2$
$=3 x+5, \quad x \geq 2 .$
Find values of $a$ for which $f(x)$ has local minima at $x=2$.

Nidhi Singhi

Numerade Educator

Problem 16

Find the polynomial of degree 6 which satisfies $\lim _{x \rightarrow 0}\left(1+\frac{J(x)}{x^{3}}\right)^{x}=e^{2}$ and has local maxima at $x=1$ and local minima at $x=0 \& x=2$.

Nidhi Singhi

Numerade Educator

Problem 17

Find greatest \& least value of the following functions in the indicated intervals:-
i. $\quad f(x)=x^{3}-3 x$ in $[0,2]$.
ii. $\quad f(x)=2 x^{3}-3 x^{2}-12 x+1$ in $\left[-2, \frac{5}{2}\right]$.
iii. $\quad f(x)=2 x^{3}-24 x+107$ in $[1,3]$.
iv. $\quad f(x)=x^{2} \ln x$ in $[1, e] .$
v. $\quad f(x)=\sqrt{\left(1-x^{2}\right)\left(1+2 x^{2}\right)}$ in $[-1,1]$.
vi. $\quad f(x)=\cos ^{-1}\left(x^{2}\right)$ in $-\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}$.
vii. $\quad f(x)=x+\sqrt{x}$ in $[0,4]$.

Carson Merrill

Numerade Educator

Problem 18

Prove that:-
i. $\quad e^{x}>1+x, \quad x \neq 0$.
ii. $\quad x-\frac{x^{3}}{6}<\sin x<x, \quad x>0$.
iii. $\frac{x}{1+x} \leq \ln (1+x) \leq x, \quad x>-1$.
iv. $\frac{x}{1+x^{2}}<\tan ^{-1} x<x, \quad x>0$.
v. $\quad \ln x>\frac{2(x-1)}{x+1}, \quad x>1$.
vi. $\quad 2 x \tan ^{-1} x \geq \ln \left(1+x^{2}\right)$.
vii. $\ln (1+x)>\frac{\tan ^{-1} x}{1+x}, \quad x>0$.
viii. $\sin x<x-\frac{x^{3}}{6}+\frac{x^{5}}{120}, \quad x>0$.
ix. $\quad \sin x+\tan x>2 x, \quad 0<x<\frac{\pi}{2}$.
x. $\quad \cosh x>1+\frac{x^{2}}{2}, \quad x \neq 0$.
xi. $1+x \ln \left(x+\sqrt{x^{2}+1}\right) \geq \sqrt{1+x^{2}}$.
xii. $\quad 2 \sin x+\tan x \geq 3 x$ for $0 \leq x<\frac{\pi}{2}$.

Aman Gupta

Numerade Educator

Problem 19

The function $f(x)=x^{4}-62 x^{2}+a x+9$ attains its maximum value on the interval $[0,2]$ at $x=1$. Find the value of $a$. =

Nidhi Singhi

Numerade Educator

Problem 20

Let $f(x)=-x^{3}+\frac{b^{3}-b^{2}+b-1}{b^{2}+3 b+2}, \quad 0 \leq x<1$
$=2 x-3, \quad 1 \leq x \leq 3 .$
Find all possible real values of $b$ such that $f(x)$ has the smallest value at $x=1$.

Nidhi Singhi

Numerade Educator

Problem 21

Use the function $f(x)=x^{\frac{1}{x}}, x>0$ to determine the bigger of the two numbers $e^{\pi} \& \pi^{e}$.

Nidhi Singhi

Numerade Educator

Problem 22

Find the intervals of concavity of the following functions:-
i. $f(x)=x^{4}+x^{3}-18 x^{2}+24 x-12$.
ii. $f(x)=3 x^{5}-5 x^{4}+3 x-2$.
iii. $f(x)=x^{6}-10 x^{4}$.
iv. $f(x)=\ln \left(x^{2}-1\right)$.
v. $f(x)=(x+1)^{4}+e^{x}$.
vi. $f(x)=x^{2} \ln x .$
vii. $f(x)=x+x^{\frac{4}{3}}$.
viii. $f(x)=x+x^{\frac{5}{3}}+1$.
ix. $f(x)=x+x^{\frac{2}{3}}$.
x. $\quad f(x)=x^{2}, \quad x \leq 0$
$=x^{3}, \quad x>0$.
$=x^{2}, \quad x>0$.
$=x^{2}, \quad x>1$.

Gregory Higby

Numerade Educator

Problem 23

Test the indicated points for point of inflection:-
i. $f(x)=x^{3}-5 x^{2}+3 x-5$ at $x=1, \frac{5}{3}, 2$.
ii. $f(x)=x^{4}-12 x^{3}+48 x^{2}$ at $x=1,2,3,4$.
iii. $f(x)=x+x^{\frac{5}{3}}-2$ at $x=0$.
iv. $f(x)=x^{2}+x^{\frac{4}{3}}+1$ at $x=0$.
v. $f(x)=x+x^{\frac{2}{3}}+4$ at $x=0$.
vi. $f(x)=x+x^{\frac{3}{5}}-3$ at $x=0$.
vii. $f(x)=\sin x+\frac{x^{3}}{3 !}-\frac{x^{5}}{5 !}$ at $x=0$.
viii. $f(x)=e^{x}-\frac{x^{2}}{2}-\frac{x^{3}}{6}$ at $x=0$.
ix. $\quad f(x)=\sin x, \quad x \geq 0$
$=x-\frac{x^{3}}{6}, \quad x<0$
at $x=0$. \{Ans. 0 is point of inflection\}

Aman Gupta

Numerade Educator

Problem 24

Test the following functions for concavity $\&$ find points of inflection:-
i. $\quad f(x)=x+36 x^{2}-2 x^{3}-x^{4}$.
ii. $\quad f(x)=3 x^{4}-8 x^{3}+6 x^{2}+12$.
iii. $f(x)=(x+2)^{6}+2 x+2$.
iv. $f(x)=\frac{x}{1+x^{2}}$.
v. $f(x)=\ln \left(1+x^{2}\right)$.
vi. $\quad f(x)=x^{4}(12 \ln x-7)$.
vii. $\quad f(x)=x \ln x$.
viii. $f(x)=\frac{\ln x}{x}$.
ix. $f(x)=x^{x}$.
x. $f(x)=x e^{x}$.

Stephanie Carter

Stephanie Carter

Numerade Educator

Problem 25

Given
$\begin{aligned} f(x) &=e^{x}, \quad x \geq 0 \\ &=a x^{3}+b x^{2}+c x+d, \quad x<0 \end{aligned}$
Find the constants $a, b, c, d$ if $f^{\prime \prime}(0)$ exists and $f(x)$ has a point of inflection at $x=-1$.

Nidhi Singhi

Numerade Educator

Problem 26

Verify Rolle's theorem for the following functions:-
i. $\quad f(x)=2 x^{3}+x^{2}-4 x-2$ in $[-\sqrt{2}, \sqrt{2}]$.
ii. $f(x)=\sin x$ in $[0, \pi]$.
iii. $f(x)=\tan x$ in $[0, \pi]$.
iv. $f(x)=\cos \frac{1}{x}$ in $[-1,1]$.
v. $\quad f(x)=x(x+3) e^{-\frac{x}{2}}$ in $[-3,0]$.
vi. $\quad f(x)=e^{x}(\sin x-\cos x)$ in $\left[\frac{\pi}{4}, \frac{5 \pi}{4}\right]$.
vii. $f(x)=|x|$ in $[-1,1]$.
viii. $f(x)=3+(x-2)^{\frac{2}{3}}$ in $[1,3]$.
ix. $\quad f(x)=\ln \left(\frac{x^{2}+a b}{(a+b) x}\right)$ in $[a, b], a>0 .$
x. $f(x)=(x-a)^{m}(x-b)^{n}$ in $[a, b]$, where $m$ and $n$ are positive integers.

Jonathon Brumley

Jonathon Brumley

Numerade Educator

Problem 27

Show that the equation $x^{3}-3 x+c=0$ cannot have two different roots in the interval $(0,1)$.

Nidhi Singhi

Numerade Educator

Problem 28

Prove that the equation $3 x^{5}+15 x-8=0$ has only one real solution.

Nidhi Singhi

Numerade Educator

Problem 29

Prove that the polynomial $x^{4}-4 x-1$ has exactly two different real roots.

Nidhi Singhi

Numerade Educator

Problem 30

Show that the equation $x e^{x}=2$ has only one solution which lies in the interval $(0,1)$.

Nidhi Singhi

Numerade Educator

Problem 31

Show that the equation $x^{4}+2 x-2=0$ has exactly one real solution in the interval $(0,1)$.

Nidhi Singhi

Numerade Educator

Problem 32

If the equation $a_{n} x^{n}+a_{n-1} x^{n-1}+\ldots \ldots+a_{1} x=0$ has a positive solution $a$, then prove that the equation $n a_{n} x^{n-1}+(n-1) a_{n-1} x^{n-2}+\ldots \ldots+a_{1}=0$ also has a positive solution which is smaller than $a$.

Nidhi Singhi

Numerade Educator

Problem 33

If $2 a+3 b+6 c=0$, then show that the equation $a x^{2}+b x+c=0$ has at least one real root between 0 and $1 .$

Nidhi Singhi

Numerade Educator

Problem 34

Let $\frac{a_{0}}{n+1}+\frac{a_{1}}{n}+\frac{a_{2}}{n-1}+\ldots \ldots \ldots+\frac{a_{n-1}}{2}+\frac{a_{n}}{1}=0 .$ Show that there exists at least one real $x$ between 0 and 1 such that $a_{0} x^{n}+a_{1} x^{n-1}+\ldots \ldots+a_{n-1} x+a_{n}=0$

Nidhi Singhi

Numerade Educator

Problem 35

Show that $f(x)=x^{2}$ satisfies Lagrange's Mean value theorem in the interval $[0,1]$ and find the value of $c$.

Nidhi Singhi

Numerade Educator

Problem 36

Prove the validity of Lagrange's theorem for the function $y=\ln x$ in the interval $[1, e]$ and find the value of $c$.

Nidhi Singhi

Numerade Educator

Problem 37

With the aid of Lagrange's theorem prove the inequalities $\frac{a-b}{a} \leq \ln \frac{a}{b} \leq \frac{a-b}{b}$, for the condition $0<b \leq a .$

Nidhi Singhi

Numerade Educator

Problem 38

With the aid of Lagrange's theorem prove the inequalities $\frac{a-b}{\cos ^{2} b} \leq \tan a-\tan b \leq \frac{a-b}{\cos ^{2} a}$, for the
condition $0<b \leq a<\frac{\pi}{2}$.

Nidhi Singhi

Numerade Educator

Problem 39

Using Mean value theorem, show that $|\cos a-\cos b| \leq|a-b|$.

Nidhi Singhi

Numerade Educator

Problem 40

Use Lagrange's theorem to prove that $1+x<e^{x}<1+x e^{x} \forall x>0$.

Nidhi Singhi

Numerade Educator

Problem 41

If $f^{\prime \prime}(x)$ exists for all points in $[a, b]$ and $\frac{f(c)-f(a)}{c-a}=\frac{f(b)-f(c)}{b-c}$, where $a<c<b$, then there is a number $\alpha$ such that $a<\alpha<b$ and $f^{\prime \prime}(\alpha)=0$.

Nidhi Singhi

Numerade Educator

Problem 42

If $f(x)$ is differentiable and $\lim _{x \rightarrow \infty} f(x)$ is finite and $\lim _{x \rightarrow \infty} f^{\prime}(x)$ is finite, then show that $\lim _{x \rightarrow \infty} f^{\prime}(x)=0$.

Melissa Munoz

Numerade Educator

Problem 43

If $f^{\prime \prime}(x) \geq 0 \forall x \in[a, b]$, show that $f\left(\frac{x_{1}+x_{2}}{2}\right) \leq \frac{f\left(x_{1}\right)+f\left(x_{2}\right)}{2}$ for $x_{1}, x_{2} \in[a, b]$.

Nidhi Singhi

Numerade Educator

Problem 44

Suppose that $f(x)$ and $g(x)$ are non-constant differentiable real valued functions on $R$. If for every $x, y \in R, f(x+y)=f(x) f(y)-g(x) g(y)$ and $g(x+y)=g(x) f(y)+f(x) g(y)$ and $f^{\prime}(0)=0$, then prove that maximum and minimum values of the function $f^{2}(x)+g^{2}(x)$ are same for all real $x$.

Nidhi Singhi

Numerade Educator