Question

Find the value of ‘a’ if f(x) = ax^2+32x+4 is continuous over [-4, 0] and differentiable over (-4, 0) and satisfy the Rolle’s theorem. Hence find the point in interval (-2,0) at which its slope of a tangent is zero

(a) 2, -2

(b) 2, -1

(c) 8, -1

(d) 8, -2

I had been asked this question in an interview for internship.

This key question is from Rolle’s Theorem in portion Differential Calculus of Engineering Mathematics

Answer 1

Right option is (d) 8, -2

For explanation I would say: Since it satisfies Rolle’s Theorem,

f’(c) = 0 = 2ac+32 ………………(1)

and, f(0) = 4 hence by Rolle’s theorem

and, f(-4) = 4 = 16a-128+4 (because f(0)=f(-4) condition of rolle’s theorem)

⇒ a = 8

from, eq.(1)

⇒ c = -2.