Question

y = x (x – 3)² decreases for the values of x given by :

A. 1 < x < 3
B. x < 0
C. x > 0
D. 0 < x < 3/2

Answer 1

Given y = x (x – 3)²

⇒ y=x(x²-6x+9)

⇒ y=x³-6x²+9x

Applying the first derivative we get

Applying the power rule we get

x=1, 3

These points divide the real number line into three intervals

(-∞, 1), (1,3) and (3,∞)

(i) in the interval (-∞, 1), f’(x)>0

∴ f(x) is increasing in (-∞,1)

(ii) in the interval (1,3), f’(x)≤0

∴ f(x) is decreasing in (1,3)

(iii) in the interval (3, ∞), f’(x)>0

∴ f(x) is increasing in (3, ∞)

Hence the interval on which the function y = x (x – 3)² is decreasing is (1,3) i.e., 1<x<3

So the correct option is option A.