fundamentals of mathematics differential calculus

Sanjay Mishra

Chapter 5 Application of Derivatives II - all with Video Answers

Educators

Section 6

SECTION-VI

03:34

Problem 1

If $f(a)=0$, then which of the following is correct
(a) $f(a+h) f^{\prime}(a-h)<0$
(b) $f(a+h) f^{\prime \prime}(a-h)>0$
(c) $f(a+h) f^{\prime \prime}(a-h)<0$
(d) $f(a+h) f^{\prime}(a-h)<0$

Aayush Gupta

Numerade Educator

02:13

If $\alpha$ and $\beta$ are two consecutive roots of $f(x)=0$, then
(a) $f^{\prime}(\gamma)=0 ; \gamma \in(\alpha, \beta)$ for some $\gamma$
(b) $f^{\prime \prime}(\gamma)=0 ; \gamma \in(\alpha, \beta)$ for some $\gamma$
(c) $f^{\prime \prime \prime}(\gamma)=0 ; \gamma \in(\alpha, \beta)$ for some $\gamma$
(d) $f^{\prime \prime \prime \prime}(\gamma)-0 ; \gamma \in(\alpha, \beta)$ for some $\gamma$

Gaurav Kalra

Numerade Educator

03:16

Problem 3

If $f^{\prime}(x) \neq 0$, then maximum number of real roots of $f^{\prime}(x)=0$ is/are
(a) no real root
(b) one
(c) two
(d) three

Pritesh Ranjan

Numerade Educator

04:54

Problem 4

If $f(x)$ is continuous in $\left[c_{1}, c_{2}\right]$ and $f^{\prime \prime}\left(c_{1}\right)-f^{\prime \prime}\left(c_{2}\right)>0$,
then minimum number of roots of $f(x)=0$ in $\left[c_{1}-1,\right.$, $\left.c_{2}+1\right]$ is
(a) 2
(b) 3
(c) 4
(d) 5

Suman Saurav Thakur

Numerade Educator

02:51

Problem 5

If $f(x)$ is continuous in $\left[c_{1} c_{2}\right]$ and $f^{\prime \prime}\left(c_{1}\right)-f^{\prime}\left(c_{2}\right)<0$ then minimum number of roots of $f(x)=0$ in $\left[c_{1}-1\right.$, $\left.c_{2}+1\right]$ is
(a) 1
(b) 2
(c) 3
(d) 4

Suman Saurav Thakur

Numerade Educator

04:54

Problem 6

If $f(x)$ is continuous in $\left[c_{1}, c_{2}\right]$ and $f^{\prime}\left(c_{1}\right)-f^{\prime}\left(c_{2}\right)>0$, then minimum number of roots of $f(x)=0$ in $\left[c_{1}-1\right.$, $\left.c_{2}+1\right]$ is
(a) 2
(b) 3
(c) 4
(d) 5

Suman Saurav Thakur

Numerade Educator

02:46

Problem 7

vIf curve has slope $1 / 6$ at $(9,0)$ then a tangent belonging to family of lines is
(a) $x+2 y-5=0$
(b) $x-2 y+3=0$
(c) $3 x-y-1=0$
(d) $3 x+y-5=0$

Donald Albin

Numerade Educator

05:21

Problem 8

A line of the family cutting positive intercepts on axes and forming triangle with coordinate axes, then minimum length of the line segment between axes is
(a) $\left(2^{2 / 3}-1\right)^{3 / 2}$
(b) $2 \sqrt{5}$
(c) $7^{3 / 2}$
(d) 27

Gaurav Kalra

Numerade Educator

06:56

Problem 9

Two perpendicular chords of curve $y^{2}-4 x-4 y+4$ $=0$ belonging to family of lines form diagonals of a quadrilateral. Minimum area of quadrilateral is
(a) 16
(b) 32
(c) 64
(d) 50

Anurag Kumar

Numerade Educator

02:19

Problem 10

The line $y=x$ meets $y=k e^{x}$ for $k \leq 0$ at
(a) no point
(b) one point
(c) two points
(d) more than two points

P Krishnamurthy

Numerade Educator

01:10

Problem 11

The positive value of $k$ for which $k e^{x}-x=0$ has only one root is
(a) $1 / \mathrm{e}$
(b) 1
(c) $e$
(d) $\log _{c} 2$

Gaurav Kalra

Numerade Educator

01:36

Problem 12

For $k>0$, the set of all values of $k$ for which $\mathrm{ke}^{x}-x=0$ has two distinct roots is
(a) $(0,1 / \mathrm{e})$
(b) $(1 / e, 1)$
(c) $(1 / c,$,
(d) $(0,1)$

Dilip Paruchuri

Numerade Educator

01:59

Problem 13

On the possible graph of $y=f(x)$ we have
(a) $x=-5$ is a point of relative minima.
(b) $x=2$ is a point of relative maxima.
(c) $x=4$ is a point of relative minima.
(d) graph of $y=f(x)$ must have a geometrical sharp corner.

Frank Rose

Numerade Educator

03:08

Problem 14

From the possible graph of $y=f(x)$, we can say that
(a) There is exactly one point of inflection on the curve.
(b) $f(x)$ increases on $-5<x<2$ and $x>4$ and decreases on $-\infty<x<-5$ and $2<x<4$.
(c) The curve is always concave down.
(d) Curve always concave up.

Nick Johnson

Numerade Educator

00:42

Problem 15

Possible graph of $y=f(x)$ is

Heather Zimmers

Numerade Educator

01:41

Problem 16

For which value of $x$ this function $f(x)=\sqrt{\left(1-x^{2}\right)\left(1+2 x^{2}\right)}$ on $[-1,1]$ possess maximum
value is
(a) $1 / 2$
(b) $1 / 4$
(c) $-1 / 2$
(d) $-1 / 4$

Hast Aggarwal

Numerade Educator

02:20

Problem 17

The largest term in the sequence $a_{n}=\left(\frac{n^{2}}{n^{3}+200}\right)$ is
(a) $\frac{59}{435}$
(b) $\frac{1}{12}$
(c) $\frac{49}{543}$
(d) $\frac{59}{434}$

Aman Gupta

Numerade Educator

01:11

Problem 18

The function $f(x)=a x+\frac{b}{x}(a, b, x>0)$, consists of two summands, one summand is proportional to the independent variable $x$, the other inversely proportional to it. Then the least value of function is at $x$ is equal to
(a) $\sqrt{a b}$
(b) $\sqrt{\frac{b}{a}}$
(c) $\sqrt{\frac{a}{b}}$
(d) $\sqrt{\frac{1}{a b}}$

Nick Johnson

Numerade Educator

02:34

Problem 19

vThe least value of the function $f(x)=\arctan x-\frac{1}{2}$. $x$ on $\left[\frac{1}{\sqrt{3}}, \sqrt{3}\right]$ is
(a) $\frac{\pi}{6}+0.25 \operatorname{In} 3$
(b) $\frac{\pi}{3}-0.25 \operatorname{In} 3$
(c) $\frac{\pi}{6}-0.25 \operatorname{In} 3$
(d) $\frac{\pi}{3}+0.25 \operatorname{In} 3$

Saurabh Chandra

Numerade Educator

02:33

Problem 20

The interval in which $f(x)$ is increasing in
(a) $\left(\frac{1}{3}, \frac{2}{3}\right)$
(b) $\left(\frac{1}{3}, \frac{1}{2}\right)$
(c) $\left(\frac{1}{3}, \frac{1}{2}\right)$ and $\left(\frac{1}{2}, \frac{2}{3}\right)$
(d) $\left(\frac{1}{3}, \frac{1}{2}\right)$ and $\left(\frac{2}{3}, 1\right)$

Gregory Higby

Numerade Educator

02:33

Problem 21

The interval in which $f(x)$ is decreasing is
(a) $\left(\frac{1}{3}, \frac{2}{3}\right)$
(b) $\left(\frac{1}{3}, \frac{1}{2}\right)$
(c) $\left(0, \frac{1}{2}\right)$ and $\left(\frac{1}{2}, \frac{2}{3}\right)$
(d) $\left(\frac{1}{3}, \frac{1}{2}\right)$ and $\left(\frac{2}{3}, 1\right)$

Gregory Higby

Numerade Educator

View

Problem 22

Let Rolle's theorem is applicable for $f(x)$ on $[a, b]$, then $a+b+c$ is (where $c$ is p.t. $f^{\prime}(x)=0$ )
(a) $2 / 3$
(b) $1 / 3$
(c) $1 / 2$
(d) $3 / 2$

Nicole Hoffman

Numerade Educator

01:13

Problem 23

Complete set of value of a for which $f(x)$ is strictly increasing for all $x \in \mathbb{R}$.
(a) $(-\infty,-5)$
(b) $(-5, \infty)$
(c) $[5, \infty)$
(d) $(-\infty, 5]$

Taylor Shimono

Numerade Educator

01:13

Problem 24

Complete set of value of a for which $f(x)$ is strictly decreasing for all $x \in \mathbb{R}$.(a) $(-\infty,-5]$
(b) $(-5, \infty)$
(c) $[5, \infty)$
(d) $(-\infty, 5]$

Taylor Shimono

Numerade Educator

03:40

Problem 25

Complete set of values of a for which $f(x)$ is onto is
(a) $(-\infty, \infty)-\{0\}$
(b) $(-5, \infty)$
(c) $[5, \infty)$
(d) None of these\begin{aligned}
&\text { 1: Let } f(x, y)=\tan ^{4} x+\tan ^{4} y+3 \cot ^{2} x \cot ^{2} y \\
&g(x, y)=3+\sin ^{2}(x+y)
\end{aligned}

Aman Gupta

Numerade Educator

02:40