Question

Find maximum and minimum value of x^3-9x^2+24x-7

Answer 1

$\large\underline{\sf{Solution-}}$

Given that

$\bf :\longmapsto\:f(x) = {x}^{3} - {9x}^{2} + 24x - 7 - - - (1)$

On differentiating both sides w. r. t. x, we get

$\sf :\longmapsto\:\dfrac{d}{dx}f(x) = \dfrac{d}{dx}( {x}^{3} - {9x}^{2} + 24x - 7)$

$\sf :\longmapsto\:f'(x) = \dfrac{d}{dx}{x}^{3} - 9\dfrac{d}{dx} {x}^{2} + 24\dfrac{d}{dx}x - \dfrac{d}{dx}7$

$\rm :\longmapsto\:f'(x) = {3x}^{2} - 18x + 24 - - - (2)$

For maxima and minima,

$\rm :\longmapsto\:Put \: f'(x) = 0$

$\rm :\longmapsto\: {3x}^{2} - 18x + 24 = 0$

$\rm :\longmapsto\: {x}^{2} - 6x + 8 = 0$

$\rm :\longmapsto\: {x}^{2} - 4x - 2x + 8 = 0$

$\rm :\longmapsto\:x(x - 4) - 2(x - 2) = 0$

$\rm :\longmapsto\:(x - 4)(x - 2) = 0$

$\bf\implies \:x = 2 \: \: \: or \: \: \: x = 4$

Now from equation (2), we have

$\rm :\longmapsto\:f'(x) = {3x}^{2} - 18x + 24$

On differentiating both sides w. r. t. x, we get

$\rm :\longmapsto\:\dfrac{d}{dx}f'(x) = \dfrac{d}{dx}( {3x}^{2} - 18x + 24 )$

$\rm :\longmapsto\:f''(x) = 6x - 18 - - - (3)$

Let we check the critical points.

Consider x = 2

$\rm :\longmapsto\:f''(2) = 6 \times 2 - 18 = 12 - 18 = - 6$

$\bf\implies \:f''(2) < 0$

$\rm :\implies\:f(x) \: is \: maximum \: at \: x = 2$

and

$\bf :\longmapsto\:Maximum \: value \: is \: f(2)$

$\rm \: = \: \: {2}^{3} - {9(2)}^{2} + 24(2) - 7$

$\rm \: = \: \: 8 - 36 + 48 - 7$

$\rm \: = \: \: 13$

Consider x = 4

$\rm :\longmapsto\:f''(4) = 6 \times 4 - 18 = 24 - 18 = 6$

$\bf\implies \:f''(4) > 0$

$\rm :\implies\:f(x) \: is \: minimum \: at \: x = 4$

and

$\bf :\longmapsto\:Minimum \: value \: is \: f(4)$

$\rm \: = \: \: {4}^{3} - {9(4)}^{2} + 24(4) - 7$

$\rm \: = \: \: 64 - 144 + 96 - 7$

$\rm \: = \: \: 9$

Basic Concepts :-

HOW TO FIND MAXIMUM AND MINIMUM VALUE OF A FUNCTION

Let given function be f(x).

Differentiate the given function, we get f'(x).

let f'(x) = 0 and find critical point say x = a

Then find the second derivative, i.e. f''(x).

Apply those critical point in the second derivative.

The function f (x) is maximum when f''(a) < 0.

The function f (x) is minimum when f''(a) > 0.