Find the points of local maxima or local minima, if any, of the following functions, using the first derivative test. Also, find the local maximum or local minimum values, as the case may be:

Question

Philoid · Accepted Answer

We have, f(x) = x³(x – 1)²

Differentiate w.r.t x, we get,

f ‘(x) = 3x²(x – 1)² + 2x³(x – 1)

= (x – 1)(3x²(x – 1) + 2x³)

= (x – 1)(3x³ – 3x² + 2x³)

= (x – 1)(5x³ – 3x²)

= x² (x – 1)(5x – 3)

For all maxima and minima,

f ’(x) = 0

= x²(x – 1)(5x – 3) = 0

= x =0, 1,

At x = f ’(x) changes from –ve to + ve

Since, x = is a point of Minima

At x =1 f ‘ (x) changes from –ve to + ve

Since, x =1 is point of maxima.

Find the points of local maxima or local minima, if any, of the following functions, using the first derivative test. Also, find the local maximum or local minimum values, as the case may be:f(x) = x3 (x – 1)2

Find the points of local maxima or local minima, if any, of the following functions, using the first derivative test. Also, find the local maximum or local minimum values, as the case may be:
f(x) = x³ (x – 1)²