How do you find the maximum, minimum and inflection points and concavity for the function #f(x)=6x^6+12x+6#?

Question

Luke Alvarez · Accepted Answer

Relative minimum at #x=(-1/3)^(1/5)#.
Always concave up and no inflection points.

#f'(x)=36x^5+12# #f''(x) =180# To find relative maxima, set #f'(x)=0#. So, #36x^5+12=0# #therefore# #x^5=-1/3# #therefore x = (-1/3)^(1/5)# Now find the function's relative minimum and maximum using the sign test. For - infinity# < x < (-1/3)^(1/5)#, the function is decreasing. For #(-1/3)^(1/5) < x< # infinity, the function is increasing. Therefore, the function has relative minimum at #x = (-1/3)^(1/5)# As for the function's concavity, we can conclude that it has no inflection points and is always concave upward because the second derivative is always positive.

Emma Aldridge · Answer

undefined To find the maximum and minimum points:

1. Take the first derivative of the function to find critical points.
2. Set the first derivative equal to zero and solve for $ x $.
3. Use the second derivative test to determine whether the critical points correspond to maximum or minimum points.

To find inflection points and concavity:

1. Take the second derivative of the function to find points of inflection.
2. Set the second derivative equal to zero and solve for $ x $.
3. Determine the concavity of the function by analyzing the sign of the second derivative in intervals between critical points.

First Derivative:
$ f'(x) = 36x^5 + 12 $

Set $ f'(x) $ equal to zero and solve for $ x $:
$ 36x^5 + 12 = 0 $
$ x^5 = -\frac{1}{3} $

Second Derivative:
$ f''(x) = 180x^4 $

Critical points:
There are no real solutions for $ x^5 = -\frac{1}{3} $, so there are no critical points.

Concavity:
Since there are no critical points, we consider the concavity of the function for all $ x $.

Since $ f''(x) = 180x^4 $ is always positive for real values of $ x $, the function is concave up for all $ x $.

There are no maximum or minimum points, and the function is concave up for all $ x $. Hence, there are no inflection points.