Question

Let $f:[2,7]\to [0,\infty )$ be a continuous and differentiable function. Then, $(f(7)-f(2))\frac{(f(7){)}^2+(f(2){)}^2+f(2)f(7)}{3}$ is, where $c\in [2,7].$

• A. $5f^2(c)f^{\prime }(c)$
• B. $5f^{\prime }(c)$
• C. $f(c)f^{\prime }(c)$
• D. None of these

Answer 1

Answer: (A)

Let $g(x)=f^3(x)$
$\Rightarrow g^{\prime }(x)=3f^2(x)\cdot f^{\prime }(x)$
$\because f:[2,7]\to [0,\infty )$
$\Rightarrow g:[2,7]\to [0,\infty )$
Using Lagrange's mean value theorem on $g(x)$, we get
$g^{\prime }(c)=\frac{g(7)-g(2)}{5},c\in [2,7]$
$\Rightarrow 5f^2(c)f^{\prime }(c)=(f(7)-f(2))\frac{(f(7){)}^2+(f(2){)}^2+f(2)f(7)}{3}$