Find greatest & least value of the following functions in...

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 $[0,4]$.

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 $[0,4]$.

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Key Concepts

Extreme Value Theorem

This theorem states that any continuous function on a closed, bounded interval will attain both a maximum and a minimum. It is fundamental in optimization problems because it guarantees that the function has both greatest and least values in the interval being considered.

Critical Points

Critical points are values in the domain of the function where the derivative is zero or undefined. They are important because a function’s local maximum or minimum can occur at these points. Finding the critical points is a key step in determining the extreme values of a function.

Derivative Analysis

Using derivatives to analyze a function involves computing the derivative, setting it equal to zero or identifying where it is undefined, and solving for the variable. This analysis helps identify critical points where the function might have turning points and potentially extreme values.

Endpoint Evaluation

Since the Extreme Value Theorem guarantees that extreme values occur on closed intervals, it is essential to evaluate the function at the endpoints of the interval. By comparing the function values at the endpoints and at the critical points, one can determine the absolute maximum and minimum values.

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