Find the points a

It is given that 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 = or x = 2

Now, for values of x close to and to the left of

f’(x) > 0.

Also, for values of x close to and to the right of , f’(x) < 0.

Then, x = is the point of local maxima.

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

Also, for values of x close to 2 and to the right of 2. 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.