Answer
Function $y$ has the absolute and local minimum value of $2$ at $x=0$. There is no absolute nor local maximum value.
Work Step by Step
$$y=e^x+e^{-x}$$
1) Find all the critical points of the function:
- Find $y'$: $$y'=e^x(x)'+e^{-x}(-x)'$$ $$y'=e^x-e^{-x}$$
- We have $y'=0$ when $$e^x=e^{-x}$$ $$x=-x$$ $$2x=0$$ $$x=0$$
- There is no value of $x$ for which $y'$ is not defined.
So $x=0$ is the only critical point of the function $y$.
2) Evaluate $y$ at the critical points:
- For $x=0$: $$y=e^0+e^{-0}=1+1=2$$
Now take a look at the graph of function $y$.
We see that function $y$ has the absolute and local minimum value of $2$ at $x=0$. There is no absolute nor local maximum value since $y$ approaches infinity constantly.
