Q. If $f:[0,1]\to \mathbb R$ is continues and differentiable. Such that $f(0)= 1$ and $[f(1)]^3 +2f(1) =5$ then prove that there exists a $c$ such that $f'(c)= 2/2+[f'(c)]^2$
How I tried to solve is that $f(0)=1$ and let $f(1)=t$ So $f'(c)= f(1)-f(0)/1-0 =t-1$
Then I tried to eliminate t from the the equation $t^3+2t-5=0$. Initially I was quite confident that it would work but after a bit it doesn't seems to..