Answer:
is a relative maxima and
is a relative minima.
Step-by-step explanation:
We have been given a function
. We are asked to find the relative extrema of the given function.
First of all, we will find first derivative of the given function as:
![f'(x)=\frac{d}{dx}(\frac{1}{8}x^3)-\frac{d}{dx}(2x)](https://tex.z-dn.net/?f=f%27%28x%29%3D%5Cfrac%7Bd%7D%7Bdx%7D%28%5Cfrac%7B1%7D%7B8%7Dx%5E3%29-%5Cfrac%7Bd%7D%7Bdx%7D%282x%29)
![f'(x)=3*\frac{1}{8}x^{3-1}-2*(x^{1-1})](https://tex.z-dn.net/?f=f%27%28x%29%3D3%2A%5Cfrac%7B1%7D%7B8%7Dx%5E%7B3-1%7D-2%2A%28x%5E%7B1-1%7D%29)
![f'(x)=\frac{3}{8}x^{2}-2*(x^0)](https://tex.z-dn.net/?f=f%27%28x%29%3D%5Cfrac%7B3%7D%7B8%7Dx%5E%7B2%7D-2%2A%28x%5E0%29)
![f'(x)=\frac{3}{8}x^{2}-2*(1)](https://tex.z-dn.net/?f=f%27%28x%29%3D%5Cfrac%7B3%7D%7B8%7Dx%5E%7B2%7D-2%2A%281%29)
![f'(x)=\frac{3}{8}x^{2}-2](https://tex.z-dn.net/?f=f%27%28x%29%3D%5Cfrac%7B3%7D%7B8%7Dx%5E%7B2%7D-2)
Now, we will find the critical points by equating derivative to 0 as:
![\frac{3}{8}x^{2}-2=0](https://tex.z-dn.net/?f=%5Cfrac%7B3%7D%7B8%7Dx%5E%7B2%7D-2%3D0)
![\frac{3}{8}x^{2}=2](https://tex.z-dn.net/?f=%5Cfrac%7B3%7D%7B8%7Dx%5E%7B2%7D%3D2)
![\frac{8}{3}*\frac{3}{8}x^{2}=\frac{8}{3}*2](https://tex.z-dn.net/?f=%5Cfrac%7B8%7D%7B3%7D%2A%5Cfrac%7B3%7D%7B8%7Dx%5E%7B2%7D%3D%5Cfrac%7B8%7D%7B3%7D%2A2)
Noe, we will check on which intervals our given function is increasing or decreasing.
![f'(-4)=\frac{3}{8}(-4)^{2}-2](https://tex.z-dn.net/?f=f%27%28-4%29%3D%5Cfrac%7B3%7D%7B8%7D%28-4%29%5E%7B2%7D-2)
![f'(-4)=\frac{3}{8}(16)-2](https://tex.z-dn.net/?f=f%27%28-4%29%3D%5Cfrac%7B3%7D%7B8%7D%2816%29-2)
![f'(-4)=3*2-2](https://tex.z-dn.net/?f=f%27%28-4%29%3D3%2A2-2)
![f'(-4)=4](https://tex.z-dn.net/?f=f%27%28-4%29%3D4)
![f'(1)=\frac{3}{8}(1)^{2}-2](https://tex.z-dn.net/?f=f%27%281%29%3D%5Cfrac%7B3%7D%7B8%7D%281%29%5E%7B2%7D-2)
![f'(1)=\frac{3}{8}-2](https://tex.z-dn.net/?f=f%27%281%29%3D%5Cfrac%7B3%7D%7B8%7D-2)
![f'(1)=-1.625](https://tex.z-dn.net/?f=f%27%281%29%3D-1.625)
![f'(4)=\frac{3}{8}(4)^{2}-2](https://tex.z-dn.net/?f=f%27%284%29%3D%5Cfrac%7B3%7D%7B8%7D%284%29%5E%7B2%7D-2)
![f'(4)=\frac{3}{8}(16)-2](https://tex.z-dn.net/?f=f%27%284%29%3D%5Cfrac%7B3%7D%7B8%7D%2816%29-2)
![f'(4)=3*2-2](https://tex.z-dn.net/?f=f%27%284%29%3D3%2A2-2)
![f'(4)=4](https://tex.z-dn.net/?f=f%27%284%29%3D4)
We know that when
, then f is increasing and when
, then f is decreasing.
Therefore,
is a relative maxima and
is a relative minima.