Question

Let f(x)=3x^3−36x+3

Input the interval(s) on which f is increasing.
Input the interval(s) on which f is decreasing.
Find the point(s) at which f achieves a local maximum.
Find the point(s) at which f achieves a local minimum.
Find the intervals on which f is concave up.
Find the intervals on which f is concave down.
Find all inflection points

Answer 1

I got that f'(x)=9x^2-36
and f"(x)=18x
so would f not be increasing from [2, infinity) and decreasing from (-infinity,2]?
I dont think there would be a local maximum but the local minimum would be (2,-45)
concave up would be [0, infinity) and concave down (-infinity, 0] with x=0 as the inflection point.
am I wrong?

Answer 2

You can tell that there will be a maximum and a minimum just from thinking about the first derivative. That is a parabola that starts out at (0,-36) and goes up on each side. That means that the first derivative WILL cross the x axis twice. Moving from left to right it will go from a positive to a negative (maximum) and then hit the bottom of the parabola before shooting back up for a negative to a positive (minimum).

The critical points are when f'(x) = 0 and therefore yes, (-2) and 2. The (-2) will be the first point I mentioned (you can graph it if you want to or do a negative/positive line (CP's line?) or just think about the equation. The -2 will be on the left and because the 9x^2 is positive, the parabola is going upwards. Therefore, -2, moving left to right, goes from positive to negative. That is your maximum. To fine the point, f(-2) = 51. So the maximum is (-2,51). The minimum is f(2).

Concavity has to do with the second derivative which you again correctly found.
f''(x)=18x.

f''(x)=0 => x= 0. => ----0---- => ---negative---0---positive---

Therefore the equation is concave down from \[[-\infty,0]\]

And concave up from
\[[0,\infty]\]

And you got that too. So I'm not sure why I did it. Oh wells.

Point of Inflection (POI) occurs when the equation continues to increase or decrease but the concavity changes.

At 0, concavity changes. At 0 the graph goes from (taken from part 1), decreasing to decreasing. So yes, f(0) is a point of inflection.

Answer 3

hmm.. I wonder why it's telling my i'm wrong then... Thank you!!