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
Now evaluate
One local minimum at (1, -54) and two
inflection points at (4, 0) and (2, -32).