Question

Let $f(x)$ be twice differentiable such that $f(x)=-f(x),f(x)=g(x),$ where $f(x)$ and $f(x)$ represent the first and second derivatives of $f(x)$ respectively. Also, if $h(x)={[f(x)]}^2+{[g(x)]}^2$ and $h(5)=5,$ then $h(10)$ is equal to:

• A. 3
• B. 10
• C. 13
• D. 5
• E. 0

Answer 1

Answer: (D)

We have, $h(x)={\{f(x)\}}^2+{\{g(x)\}}^2$ On differentiating w.r.t. $x,$ we get $\Rightarrow$ $h(x)=2f(x)f(x)+2g(x)g(x)$ ??(i) Now, $f(x)=g(x)$ and $f(x)=-f(x)$ $\Rightarrow$ $f(x)=g(x)$ and $f(x)=-f(x)$ $\Rightarrow$ $-f(x)=g(x)$ Thus, $f(x)=g(x)$ and $g(x)=-f(x)$ From (i) $h(x)=-2g(x)g(x)+2g(x)g(x)$ $=0$ $\Rightarrow$ $h(x)=5$ $[\because h(5)=5]$ $\therefore$ $h(10)=5$