Never expected Rolle's

Calculus Level pending

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 \(k\)?

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