Question

Find the value of c if f(x) = sin^3(x)cos(x), is continuous over interval [0, ^π⁄2] and differentiable over interval (0, ^π⁄2) and c ∈(0, ^π⁄2)

(a) 0

(b) ^π⁄6

(c) ^π⁄3

(d) ^π⁄2

I had been asked this question in unit test.

My doubt is from Rolle’s Theorem topic in chapter Differential Calculus of Engineering Mathematics

Answer 1

The correct option is (c) ^π⁄3

Easiest explanation: f(x) = sin^3(x)cos(x)

f(0) = 0

f(^π⁄2) = 0

Hence,

f’(c) = 3sin^2(c)cos(c)cos(c) – sin^4(c) = 0

3sin^2(c)cos^2(c) – sin^4(c) = 0

sin^2(c)[3 cos^2(c) – sin^2(c)] = 0

either, sin^2(c)=0 or 3 cos^2(c) – sin^2(c) = 0

Since sin^2(c) cannot be zero because c cannot be 0

Hence, 3 cos^2(c) – sin^2(c)=0

tan^2(c) = 3

tan(c) = √3

c = ^π⁄3.