Question

Which of the following is not a necessary condition for Cauchy’s Mean Value Theorem?

(a) The functions, f(x) and g(x) be continuous in [a, b]

(b) The derivation of g'(x) be equal to 0

(c) The functions f(x) and g(x) be derivable in (a, b)

(d) There exists a value c Є (a, b) such that, \(\frac{f(b)-f(a)}{g(b)-g(a)} = \frac{f'(c)}{g'(c)}\)

The question was asked in an international level competition.

The question is from Cauchy’s  Mean Value Theorem in division Differential Calculus of Engineering Mathematics

Answer 1

Correct option is (b) The derivation of g'(x) be equal to 0

To elaborate: Cauchy’s Mean Value theorem is given by, \(\frac{f(b)-f(a)}{g(b)-g(a)} = \frac{f'(c)}{g'(c)}\), where f(x) and g(x) be two functions which are derivable in [a, b] and g'(x)≠0 for any value of x in [a, b] and where c Є (a, b).