# How do you find the maximum, minimum and inflection points and concavity for the function #F(x) = 2x(x-4)^3#?

##### 1 Answer

Local minimum: (1, -54)

Inflection Points: (4, 0) and (2, -32)

This function has no local maximum.

graph{2x(x-4)^3 [-1.47, 6.0, -60.732, 15.714]}

#### Explanation:

**Find local extrema**

You can find the local extrema (maximum and minimum) by setting the first derivative to zero.

To find the first derivative, directly apply the product rule:

and then the chain rule along with the power rule:

Factor out

Now set the first derivative to zero and solve for

By the factor theorem, we have

**Verify local extrema**

Still, we aren't 100% sure if these x-coordinates corresponds to extrema; nor do we know whether each of them is a local maximum or a local minimum. To be safe (and to avoid potential pitfalls) we need to apply the second derivative test.

Differentiate the first derivative to find

Again, apply the product rule, the chain rule, and the power rule:

Factor out

Now plug zeros of the first derivative into the second derivative. Depending on the sign of the value (or the output's relationship with zero, to be precise,) there is going to be a

- Local Maximum if
#F''(x)>0# - Local Minimum if
#F''(x)<0# - Point of inflection if
#F''(x)=0#

In this case we have

**Find** (other) **points of inflections**

So now we've found a point of inflection. Still, we need to set the second derivative to zero and solve for

Again, apply the factor theorem, and we find

**Verify points of inflections**

graph{(x-2)(x-4) [0.391, 5.39, -1.48, 1.02]}

Important: Make sure that the *crosses* rather than touches the

Now evaluate

One local minimum at (1, -54) and two

inflection points at (4, 0) and (2, -32).