Question

f(x)= 10X³-15 x²+10 find maxima and minima​

Answer 1

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

Given that,

$\rm :\longmapsto\:f(x) = {10x}^{3} - {15x}^{2} + 10 - - - (1)$

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

$\rm :\longmapsto\: \: \dfrac{d}{dx} f(x) = \dfrac{d}{dx}({10x}^{3} - {15x}^{2} + 10)$

$\rm :\longmapsto\: \: f'(x) = \dfrac{d}{dx}{10x}^{3} - \dfrac{d}{dx}{15x}^{2} + \dfrac{d}{dx}10$

$\rm :\longmapsto\: \: f'(x) = 10\dfrac{d}{dx}{x}^{3} - 15 \dfrac{d}{dx}{x}^{2} + \dfrac{d}{dx}10$

$\: \: \: \: \: \: \: \: \: \: \: \: \red{ \boxed{ \because \: \bf \: \dfrac{d}{dx}k \: f(x) = k \: \dfrac{d}{dx}f(x)}}$

$\rm :\longmapsto\:f'(x) = {30x}^{2} - 30x - - - (2$

$\: \: \: \: \: \: \: \: \: \: \: \: \red{ \boxed{ \because \: \bf \: \dfrac{d}{dx} {x}^{n} = {nx}^{n - 1} }}$

For maxima or minima,

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

$\rm :\longmapsto\: {30x}^{2} - 30x = 0$

$\rm :\longmapsto\:30x(x - 1) = 0$

$\rm :\implies\:x = 0 \: \: or \: \: x = 1$

Now, On differentiating equation (2), w . r. t. x, we get

$\rm :\longmapsto\:f''(x) = 60x - 30$

Now, we have to check the points x = 0 and x = 1.

Consider,

$\rm :\longmapsto\:When \: x = 0$

$\rm :\longmapsto\:f''(0) = 60 \times 0 - 30$

$\rm :\longmapsto\:f''(0) = - 30$

$\rm :\longmapsto\:f''(0) < 0$

$\bf\implies \:f(x) \: is \: maximum \: at \: x = 0$

and

$\rm :\longmapsto\:Maximum \: value \: = f(0) = 10$

Consider,

$\rm :\longmapsto\:When \: x = 1$

$\rm :\longmapsto\:f''(1) = 60 \times 1- 30$

$\rm :\longmapsto\:f''(1) = 60 - 30$

$\rm :\longmapsto\:f''(1) = 30$

$\rm :\longmapsto\:f''(1) > 0$

$\bf\implies \:f(x) \: is \: minimum \: at \: x = 1$

and

$\rm :\longmapsto\:Minimum \: value \: = f(1) = 10 - 15 + 10 = 5$

Basic concept used :-

HOW TO FIND MAXIMUM AND MINIMUM VALUE OF A FUNCTION

Let given function be f(x).

• Differentiate the given function to get f'(x)

• let f'(x) = 0 and find critical points.

• Then find the second derivative f''(x).

• Apply these critical points in the second derivative.

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

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