Question

Let $f : [a, b] \to \, R $ be such $f$ is differentiable in $(a, b), f$ is continuous at $x = a \& x = b$ and moreover $f(a) = 0 = f(b)$. Then

• A. there exists at least one point c in (a, b) such that f'(c) = f(c)
• B. f'(x) = f(x) does not hold at any poit in (a, b)
• C. at every point of (a, b), f'(x) > f(x)
• D. at every point of (a, b), f'(x) < f(x)

Answer 1

Answer: (A)

Let $g(x)=e^{-x} f(x)$
such that $ g(a)=0, g(b)=0$
and $g(x)$ is continuous and differentiable.
Then, for atleast one value of $c \in(a, b)$ such that $g(c)=0$
Now, $g'(x)=e^{-x} f'(x)+\left(-e^{-x}\right) f(x)$
$\Rightarrow g'(c)=e^{-c} f^{\prime}(c)+\left(-e^{-c}\right) f(c)=0$
$\Rightarrow e^{-c} f'(c)=e^{-c} f(c) $
$\Rightarrow f^{\prime}(c)=f(c)$