Question

f(x) = \(\frac{sin(x)}{x}\), How many points exist such that f'(c) = 0 in the interval [0, 18π].

(a) 18

(b) 17

(c) 8

(d) 9

I have been asked this question in homework.

This interesting question is from Rolle’s Theorem in section Differential Calculus of Engineering Mathematics

Answer 1

The correct option is (a) 18

The best explanation: We have the sine function that takes the value of zero at Integral multiples of π

But for \(\frac{sin(x)}{x}\) we have the exceptional value of \(lim_{x \rightarrow 0}\frac{sin(x)}{x}\) reaching one.

So leaving the first interval [0, π], for every other interval of the form [nπ (n + 1)π] we must have f(nπ) = f((n + 1)π)

By Rolles theorem we have

f’ (c) = 0 For every interval of the form [nπ (n + 1)π]

There are 17 such intervals.