Let be continuous and differentiable function and if where

MATHEMATICS · Subjective · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation

Problem

Let

f:[1,3]\rightarrow[0,\infty)

be continuous and differentiable function and if

(f(3)-f(1))\cdot(f^2(3)+f^2(1)+f(3)f(1))=kf^2(c)f'(c)

where

c\in(1,3)

, then find the value of

k