Never expected Rolle's

Calculus Level 3

Let $f(x)$ be a continuous and differentiable function on $[0,1]$ , such that $f(0)\ne0$ and $f(1)=0$ . We can conclude that there exists $c\in (0,1)$ such that

$cf'(c)+f(c)=k$

What is the value of $k$ ?

The answer is 0.

1 solution

Otto Bretscher
Sep 22, 2015

Define $g(x)=xf(x)$ , with $g'(x)=xf'(x)+f(x)$ , and $g(0)=g(1)=0$ . Rolle's theorem tells us that there exists a $c$ on $(0,1)$ such that $0=g'(c)=cf'(c)+f(c).$

Never expected Rolle's

The answer is 0.

1 solution

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