Question

Let $f(x)$ and $g(x)$ be two differentiable functions such that $f'(x)=g(x)$ and $g'(x)=-f(x)$. Let $h(x)=(f(x))^2+(g(x))^2$ and $h(3)=100$. Then $h(100)$ is equal to

• A. $100$
• B. $10$
• C. $50$
• D. $200$
• E. $300$

Hint:

If derivative is zero, function is constant.

Answer 1

The Correct Option is A

Concept:
• Check derivative of $h(x)$ to see if constant

Step 1: Differentiate $h(x)$
\[ h(x) = f^2 + g^2 \] \[ h'(x) = 2f f' + 2g g' \]

Step 2: Substitute given values
\[ = 2f(g) + 2g(-f) = 2fg - 2fg = 0 \]

Step 3: Conclusion
\[ h'(x) = 0 \Rightarrow h(x) = \text{constant} \] \[ h(100) = h(3) = 100 \] Final Conclusion:
\[ = 100 \]