# What are the relative extrema of this equation? x^4 - 2x^3

##### 2 Answers

Local minimum at

#### Explanation:

To find local extrema, we use the first an second derivative tests.

For the sake of the first derivative test, let's factor this to:

If we set

Now we use the second derivative test to determine minimum/maximum/point of inflection:

Factoring, we get

Now we evaluate our two possible extrema using the second derivative test:

All of this can be observed on the graph of the original function:

graph{x^4-2x^3 [-1.49, 4.67, -1.958, 1.122]}

We have a local minimum at

#### Explanation:

Determine

We have prospective extrema at

At each of these intervals, we want to determine if **If #f'(x)>0# on an interval, #f(x)# is increasing on that interval. If #f'(x)<0# on an interval, #f(x)# is decreasing on that interval.**

We have an extremum at any

Let's select

Let's select

Let's select

**Since #f(x)# went from decreasing on #(0,3/2)# to increasing on #(3/2,∞),# we have a local minimum at #x=3/2.#**

To determine the

We have a local minimum at