Question

Attached is an image of what I am struggling with

Answer 1

The Solution:

The derivative of the function f(x) is given as

Step 2:

We shall obtain the function f(x) by integrating the derivative f'(x) with respect to x.

`\int f^(\prime)(x)dx=\int \frac{-2}{3x^{(1)/(3)}}dx=\int (-2)/(3)x^{-(1)/(3)}dx=-(2)/(3)\int x^{-(1)/(3)}dx`
`\begin{gathered} -(2)/(3)\int x^{-(1)/(3)}dx=((-(2)/(3)x)/((2)/(3))^{-(1)/(3)+1})+C=-x^{(2)/(3)}+C \\ \text{Where C is the constant of integration.} \\ \text{ So,} \\ f(x)=-x^{(2)/(3)}\text{ at the origin ( i.e at (0,0)} \end{gathered}`

Using Desmos graph plotter, we have the graph below:

Clearly from the graph above, we have that:

The function f(x) is increasing on the interval

`(-\infty,0)`

The function f(x) is decreasing on the interval

`(0,\infty)`
`\begin{gathered} \text{ We shall test the relative extrema of f(x) by substituting 0 for x in f(x).} \\ f(x)=-x^{(2)/(3)} \\ f(0)=-(0)^{(2)/(3)}=0 \end{gathered}`

The relative extrema occur at (0,0) meaning the value of x at the point is 0, and it is a Global maximum type of relative extrema because f(x) has its greatest value at the point.