Question

Let ƒ be a differentiable function on R and

h(x) = ƒ(x) – (ƒ(x))² + (ƒ(x))³ for all x ∈ R. Then-

• A. H increases whenever ƒ decreases
• B. H decreases whenever ƒ increases
• C. H increases or decreases according as f increases or decreases
• D. Nothing can be claimed in general

Answer 1

Answer: (C)

H increases or decreases according as f increases or

decreases

Answer 2

h′(x) = ƒ′(x) – 2ƒ(x) ƒ′(x) + 3(ƒ(x))² ƒ′(x) = ƒ′ (x)

[1 – 2ƒ(x) + 3(ƒ(x))²]

Since 1 – 2ƒ(x) + 3(ƒ(x))²

= 3$\left\lbrack \left( ƒ(x) - \frac{1}{3} \right)^{2} + \frac{2}{9} \right\rbrack$ > 0 for all x so h′(x) > 0 or < 0

according as ƒ′(x) > 0 or < 0. Hence h increases or decreases according as ƒ increases or decreases.