Derivative based question

Question

If f(x) is a function such that f(x) + f''(x)=0 and g(x)= (f(x))²+(f'(x))² and g(3)= 8, then g(8)= ?

Answer 1

g(x) = [f(x)]² + [f'(x)]² and f(x) + f"(x) = 0 and g(3) = 8

 

By the chain rule, g'(x) = 2[f(x)]f'(x) + 2[f'(x)]f"(x)

 

                                  = 2f'(x)[f(x) + f"(x)] = 2f'(x)(0) = 0

 

Since g'(x) = 0, g(x) is a constant function.

 

But, g(3) = 8.

 

So, g(8) = 8

 

Answer 2

f(x) + f''(x) = 0

 

This is a differential equation whose general solution is

 

f(x) = a sinx + b cosx, 

 

where a and b are constants to be determined from boundary conditions.

 

f'(x) = a cosx - b sinx

 

g(x) = f(x)² + f'(x)² = (a sinx + b cosx)² + (a cosx - b sinx)²

g(x) = a²sin²x + 2ab sinx cosx + b² cos²x + a²cos²x - 2ab cosx sinx + b²sin²x

g(x) = a²(sin²x + cos²x) + b²(cos²x + sin²x) = a² + b²

 

g(x) = a² + b² = constant ⇒ g(8) = g(3) = 8