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, $g^{\prime }(x)=e^{-x}{f}^{\prime }(x)+\left(-e^{-x}\right)f(x)$
$\Rightarrow g^{\prime }(c)=e^{-c}{f}^{\prime }(c)+\left(-e^{-c}\right)f(c)=0$
$\Rightarrow e^{-c}{f}^{\prime }(c)=e^{-c}f(c)$
$\Rightarrow f^{\prime }(c)=f(c)$