Question

If $f:[-5,5]\to R$ is a differentiable function and if $f\left(x\right)$ does not vanish anywhere, then prove that $\int (-5)\ne \int \left(5\right)$.

Answer 1

If we suppose that f(-5) = f(5), then f would satisfy all the conditions of Rolle's theorem on [-5,5].

$(\because$ Differentiable function is always continuous )

Hence, there would exist atleast. one c $ϵ(-5,5)$ such that f'(c)=0

But f' (x) does not vanish any where, therefore, our supposition is wrong and $f\left(5\right)\ne f(-5)$.