Let $f$ be a continuous function on the closed interval $[a,b]$ and differentiable on the open interval $(a,b),$ where $a<b.$ It is known that there exists $c$ in $(a,b)$ such that
$$f^{‘}(c)= \frac{f(b)-f(a)}{b-a}$$
Determine $c$ for $f(x)=2x^3-3x^2+7x-2$ on [a,b] with $a=1$ and $b=6.$
