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 (ii) point of inflexion

Answer 1

concopts :-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.