If f(x) is a twice differentiable function such that f"(x)=-f(x), f'(x)=g(x), h(x)=f2(x)+g2(x) and h(10)=10, then h(5) is equal to

Question

If f(x) is a twice differentiable function such that f"(x)=-f(x), f'(x)=g(x), h(x)=f2(x)+g2(x) and h(10)=10, then h(5) is equal to (A) 5 (B) 15 (C) 10 (D) 17

Tardigrade Admin · Accepted Answer

h'(x)=2f⋅ f'+2gg' =2f⋅ f'+2(f')(f")(.∵ g=f'.) =2f⋅ f'+2f'(- f)=0 ∴ h'(x)=0⇒ h(x) is a constant function ∴ h(x)=10⇒ h(5)=10

Q. If $f (x)$ is a twice differentiable function such that $f^{"} (x) = - f (x), f^{^{'}} (x) = g (x),$ $h (x) = f^{2} (x) + g^{2} (x)$ and $h (10) = 10,$ then $h (5)$ is equal to

$5$

$15$

$10$

$17$

$h^{^{'}} (x) = 2 f \cdot f^{^{'}} + 2 g g^{^{'}}$
$= 2 f \cdot f^{^{'}} + 2 (f^{^{'}}) (f^{"}) (∵ g = f^{^{'}})$
$= 2 f \cdot f^{^{'}} + 2 f^{^{'}} (- f) = 0$
$∴ h^{^{'}} (x) = 0 \Rightarrow h (x)$ is a constant function
$∴ h (x) = 10 \Rightarrow h (5) = 10$

Q. If f(x) is a twice differentiable function such that f"(x)=−f(x),f′(x)=g(x), h(x)=f2(x)+g2(x) and h(10)=10, then h(5) is equal to

5

15

10

17

h′(x)=2f⋅f′+2gg′ =2f⋅f′+2(f′)(f")(∵g=f′) =2f⋅f′+2f′(−f)=0 ∴h′(x)=0⇒h(x) is a constant function ∴h(x)=10⇒h(5)=10

Q. If $f (x)$ is a twice differentiable function such that $f^{"} (x) = - f (x), f^{^{'}} (x) = g (x),$ $h (x) = f^{2} (x) + g^{2} (x)$ and $h (10) = 10,$ then $h (5)$ is equal to

$5$

$15$

$10$

$17$

$h^{^{'}} (x) = 2 f \cdot f^{^{'}} + 2 g g^{^{'}}$
$= 2 f \cdot f^{^{'}} + 2 (f^{^{'}}) (f^{"}) (∵ g = f^{^{'}})$
$= 2 f \cdot f^{^{'}} + 2 f^{^{'}} (- f) = 0$
$∴ h^{^{'}} (x) = 0 \Rightarrow h (x)$ is a constant function
$∴ h (x) = 10 \Rightarrow h (5) = 10$