Question

Let $f$ be a twice differentiable function such that $g\text{'}\left(x\right)=-f\left(x\right)$ and $f\text{'}\left(x\right)=g\left(x\right)$, $h\left(x\right)={\left\{f\left(x\right)\right\}}^2+{\left\{g\left(x\right)\right\}}^2$. If $h\left(5\right)=11$, then $h\left(10\right)$ is equal to

• A. $22$
• B. $11$
• C. $0$
• D. $20$

Answer 1

The correct option is B

$11$

Determine the value of $h\left(10\right)$.

Consider the given equation as: $h\left(x\right)={\left\{f\left(x\right)\right\}}^2+{\left\{g\left(x\right)\right\}}^2$

Then, $h\text{'}\left(x\right)=2f\left(x\right).f\text{'}\left(x\right)+2g\left(x\right).g\text{'}\left(x\right)......\left(1\right)$

From the given data

$$\begin{gathered} g\left(x\right)=f\text{'}\left(x\right) \\ g\text{'}\left(x\right)=f\text{'}\text{'}\left(x\right) \\ f\text{'}\text{'}\left(x\right)=-f\left(x\right) \end{gathered}$$

Substitute the above values in the Equation$\left(1\right)$

$$\begin{gathered} h\text{'}\left(x\right)=2f\left(x\right).f\text{'}\left(x\right)-2g\left(x\right).f\left(x\right) \\ h\text{'}\left(x\right)=2f\left(x\right).g\left(x\right)-2g\left(x\right).f\left(x\right) \\ h\text{'}\left(x\right)=0 \end{gathered}$$

Since, $h\left(x\right)$ has some constant value.

Then, the value of $h\left(10\right)$ is given as:

$$\begin{gathered} h\left(10\right)=h\left(5\right)=11 \\ h\left(10\right)=11 \end{gathered}$$

Hence, The correct answer is Option B