Question

Find the points at which the function f given by f (x) = (x – 2)⁴ (x + 1)³ has
(i) local maxima (ii) local minima
(iii) point of inflexion

Answer 1

Concepts :-1. when f'(a) > 0 for x belongs to the left side of a

and f'(a) < 0 for x belongs to the right side of a.

then, at x = a , function attains local maxima .

2. when f'(a) > 0 for x belongs to right side of a

and f'(a) < 0 for x belongs to left side of a.

then, at x = a , function attains local minima.

3. when the value of x varies through a , f’(a) does not changes its sign. then at x = a , is the point of inflexion.

function is f (x) = (x – 2)⁴ (x + 1)³

⇒ f’(x) = 4(x - 2)³ (x + 1)³ + 3(x + 1)²(x - 2)⁴

=(x - 2)³(x + 1)²[4(x + 1) + 3 (x - 2)]

=(x - 2)³(x + 1)²(7x - 2)

Now, f’(x) =0

⇒ x = - 1 and x = 2/7 or x = 2

Now, for values of x close to 2/7 and to the left of 2/7. we get , f’(x) > 0.

Also, for values of x close to 2/7 and to the right of 2/7. we get, f’(x) < 0.

Then, x = 2/7 is the point of local maxima.

Now, for values of x close to 2 and to the left of 2. we get, f’(x) < 0.

Also, for values of x close to 2 and to the right of 2. we get , f’(x) > 0.

Then, x = 2 is the point of local minima.

Now, as the value of x varies through - 1, f’(x) does not changes its sign.

Then, x = - 1 is the point of inflexion.