Question

Let f(x) be a func. satisfying f(-x)=f(x) for all real x.if f"(x) exist, find its value.

Answer 1

`f''(x)=f''(-x)`

Explanation:

A function satisfying the equation
`f(x)=f(-x)` is said to be an even function. This denomination comes from the fact that the same relation is satisfied for functions of the form
`x^(n)` with
`n` even. Observe that if
`f` is twice differentiable we can derivate using the chaing rule as follows:

`f(x)=f(-x)` implies
`f'(x)=f'(-x)\cdot (-1)=-f'(-x)`

Applying the chain rule again we have:

`f'(x)=-f'(-x)` implies
`f''(x)=-f''(-x)\cdot (-1)=f''(-x)`

So we have that function
`f''(x)` is also an even function.