Question

Find the value of ‘a’ & ‘b’ if f(x) = ax^2 + bx + sin(x) is continuous over [0, π] and differentiable over (0, π) and satisfy the Rolle’s theorem at point c = ^π⁄4.

(a) 0.45,1.414

(b) 0.45,-1.414

(c) -0.45,1.414

(d) -0.45,-1.414

The question was asked in final exam.

I'd like to ask this question from Rolle’s Theorem in chapter Differential Calculus of Engineering Mathematics

Answer 1

Right choice is (b) 0.45,-1.414

Easy explanation: Since function f(x) is continuous over [0,π] and satisfy rolle’s theorem,

⇒ f(0) = f(π) = 0

⇒ f(π) = a π^2 + b π=0

⇒ a π+b=0 ………………….(1)

Since it satisfies rolle’s theorem at c = ^π⁄4

f’(c) = 2ac + b + Cos(c) = 0

⇒ a(^π⁄2) + b + ^1⁄√2 = 0 ………………..(2)

From eq(1) and eq(2) we get,

⇒ a = 0.45

⇒ b = -1.414.