# Over the x-value interval #[-10, 10]#, what are the local extrema of #f(x) = x^3#?

##### 1 Answer

- Find the derivative of the given function.
- Set the
**derivative equal to 0**to find the critical points. **Also use the endpoints as critical points**.

4a. Evaluate the original function using **each** critical point as an input value.

OR

4b. Create a **sign table/chart** using **values between the critical points** and record their **signs** .

5.Based on the results from STEP 4a or 4b determine if each of the criticals points are a **maximum** or a **minimum** or an **inflections** points.

**Maximum** are indicated by a **positive** value, followed by the **critical** point, followed by a **negative** value.

**Minimum** are indicated by a **negative** value, followed by the **critical** point, followed by a **positive** value.

**Inflections** are indicated by a **negative** value, followed by the **critical** point, followed by **negative** OR a **positive** value, followed by the **critical** point, followed by **positive** value.

**STEP 1:**

**STEP 2:**

**STEP 3:**

**STEP 4:**

**Point (-10,-1000)**

**Point (0,0)**

**Point (-10,1000)**

**STEP 5:**

Because the result of f(-10) is the smallest at -1000 it is the minimum.

Because the result of f(10) is the largest at 1000 it is the maximum.

f(0) has to be an inflection point.

OR

Check of my work using a signs chart

The **critical point** of **positive** values so it is an **inflection** point.

**Point (-10,-1000)**

**Point (0,0)**

**Point (-10,1000)**