What will be the value of x for which the value of cosx is minimum?

(a) 0

(b) -1

(c) 1

(d) Cannot be determined

I have been asked this question in an international level competition.

This question is from Calculus Application topic in chapter Application of Calculus of Mathematics – Class 12

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Correct option is (b) -1

Easiest explanation: Let, f(x) = cosx

Then, f’(x) = -sinx and f”(x) = -cosx

At an extreme point of f(x) we must have,

f’(x) = 0

Or -sinx = 0

Or x = nπ where, n – any integer.

If n is an odd integer i.e., n = 2m + 1 where m is any integer, then at,

x = (2m + 1)π we have, f”(x) = [(2m + 1)π] = -cos(2mπ + π) = -cosπ = -1(-1) = 1

So, f”(x) is positive at x = (2m + 1)π

Hence, f(x) = cosx is minimum at x = (2m + 1)π.

So, the minimum value of cosx is cos(2mπ + π) = cosπ = -1.