= -f(x) $ suggests a trigonometric solution, typically of the form:

$$ f(x) = A \cos(x) + B \sin(x) $$

where $ A $ and $ B $ are constants.

Relation of $ g(x) $: Since $ g(x) = f^{\prime}(x) $, we differentiate $ f(x) $:

$$ f^{\prime}(x) = -A \sin(x) + B \cos(x) $$

Therefore, we have:

$$ g(x) = -A \sin(x) + B \cos(x) $$

Expression for $ F(x) $: We need to evaluate $ F(x) $:

$$ F(x) = \left( f\left( \frac{x}{2} \right) \right)^2 + \left( g\left( \frac{x}{2} \right) \right)^2 $$

Evaluate at $ x = 5 $: We know $ F(5) = 5 $:

Substitute $ \frac{x}{2} = \frac{5}{2} $:

$$ F(5) = \left( A \cos\left( \frac{5}{2} \right) + B \sin\left( \frac{5}{2} \right) \right)^2 + \left( -A \sin\left( \frac{5}{2} \right) + B \cos\left( \frac{5}{2} \right) \right)^2 = 5 $$

This implies:

$$ A^2 \left( \cos^2\left( \frac{5}{2} \right) + \sin^2\left( \frac{5}{2} \right) \right) + B^2 \left( \sin^2\left( \frac{5}{2} \right) + \cos^2\left( \frac{5}{2} \right) \right) = 5 $$

Simplifying using $ \cos^2(x) + \sin^2(x) = 1 $:

$$ A^2 + B^2 = 5 $$

Determine $ F(10) $: Now evaluate $ F(10) $:

Substitute $ \frac{x}{2} = \frac{10}{2} = 5 $:

$$ F(10) = \left( A \cos(5) + B \sin(5) \right)^2 + \left( -A \sin(5) + B \cos(5) \right)^2 $$

As before, we simplifying yields:

$$ A^2 + B^2 = 5 $$

Therefore, $ F(10) = 5 $.

Thus, the value of $ F(10) $ is $\boxed{5}$.

Differentiation

Continue this paper

Similar Differentiation PYQs

More practice from this question