Question

34 Let $f :[1,3] \rightarrow[0, \infty)$ be continuous and differentiable function and if $(f(3)-f(1)) \cdot\left(f^2(3)+f^2(1)+f(3) f(1)\right)=k f^2(c) f^{\prime}(c)$ where $c \in(1,3)$, then find the value of $k$.

Answer 1

Let $F(x)=f^3(x)$
and $F(x)$ is continuous and differentiable function in $[1,3]$.
$\therefore \frac{F(3)-F(1)}{3-1}=F^{\prime}(c) \text { (using L.M.V.T.) }$
$\frac{f^3(3)-f^3(1)}{2}=3 f^2(c) \cdot f^{\prime}(c) \Rightarrow k=6 $