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
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.