Verify Rolle’s theorem for each of the following functions on indicated intervals :
f(x) = sin x + cos x – on open square brackets 0 comma straight pi over 2 close square brackets

Asked by Topperlearning User | 4th Aug, 2014, 04:16: PM

Expert Answer:

Since sin x and cos x are everywhere continuous and differentiable. Therefore, f (x) = sin x + cos x – 1 is continuous on open square brackets 0 comma straight pi over 2 close square brackets and differentiable on open parentheses 0 comma pi over 2 close parentheses

Also, f (0) = sin 0 + cos 0 – 1 = 0 and f open parentheses pi over 2 close parentheses equals sin space pi over 2 plus cos space pi over 2 minus 1 equals 1 minus 1 equals 0
\                 f (0) = f open parentheses pi over 2 close parentheses
Thus, f (x) satisfied conditions of Rolle’s theorem on open square brackets 0 comma straight pi over 2 close square brackets Therefore, there exists c ε open parentheses 0 comma pi over 2 close parentheses such that f ¢ (c) = 0
Þ cos x – sin x = 0 Þ sin x = cos x Þ tan x = 1 Þ x = straight pi over 4
Thus, c = pi over 4 element of open parentheses 0 comma pi over 2 close parentheses such that f ¢ (c) = 0
Hence, Rolle’s theorem is verified

Answered by  | 4th Aug, 2014, 06:16: PM