Question

If$f\text{'}\left(x\right)=g\left(x\right)$ and $g\text{'}\left(x\right)=-f\left(x\right)$ for all $x$ and $f\left(5\right)=2=f\text{'}\left(5\right)$. Then $f^2\left(10\right)+g^2\left(10\right)$ is

• A. $8$
• B. $16$
• C. $32$
• D. $64$

Answer 1

The correct option is A

$8$

Explanation for the correct option:

Step 1: Find the first derivative of $[f{\left(x\right)}^2+g{\left(x\right)}^2]$

Let us consider,

$h\left(x\right)=[f{\left(x\right)}^2+g{\left(x\right)}^2]$

Differentiating $h\left(x\right)$ with respect to $x$ we get,

$\begin{array}{rcl}h\text{'}\left(x\right)& =& 2f\left(x\right)f\text{'}\left(x\right)+2g\left(x\right)g\text{'}\left(x\right)\\ & =& 2f\left(x\right)g\left(x\right)+2g\left(x\right)[-f(x\left)\right]\\ & =& 0\end{array}$

Step 2: Find the value of $f^2\left(10\right)+g^2\left(10\right)$

Finding $h\left(10\right)$ we will get,

$$\begin{gathered} h\text{'}\left(x\right)=0 \\ \Rightarrow h\left(x\right)=cons\tan t \\ \Rightarrow h\left(5\right)=h\left(10\right) \end{gathered}$$

Now,

$\begin{array}{rcl}{f}^2\left(5\right)+g^2\left(5\right)& =& f^2\left(5\right)+f{\text{'}}^2\left(5\right)\\ & =& 2^2+2^2\\ & =& 8.\end{array}$

Hence, the correct option is A.