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

Answer:

$f''(x)=f''(-x)$

Step-by-step 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.