# 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

**Solution:**

The given function is f (x) = (x - 2)^{4} (x + 1)^{3}

Thereofore,

On differentiating wrt x, we get

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, x = 2/7, x = 2

For values of x close to 2/7 and to the left of 2/7, f' (x) > 0

Also, for values of x close to 2/7 and to the right of 2/7, f' (x) > 0

Thus, x = 2/7 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

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

Now, as the value of x varies through - 1,

f' (x) does not change its sign.

Thus, x = - 1 is the point of inflexion

NCERT Solutions Class 12 Maths - Chapter 6 Exercise ME Question 13

## 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

**Summary:**

f given by f (x) = (x - 2)^{4} (x + 1)^{3}. x = 2/7 is the point of local maxima. x = 2 is the point of local minima. x = - 1 is the point of inflection

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