For f(x) = 2x4 - 4x2 + 1 find the open intervals in which the function is increasing and decreasing.

Question

Find open intervals where the function is concave up and concave down.  

Sketch the graph the function - label all local maximums, all local minimums, and any inflection points.

Answer 1

Hey! I'll start by answering the first part. To find increasing or decreasing, we want to find the first derivative, or df/dx = 8x^3 - 8x. This has zeroes at 0, 1 and -1. Intervals where df/dx (interpreted as the rate of change of the function) is positive indicate the function is increasing, negative correspond to decreasing. Therefore, the function is increasing on the intervals (-1,1), (1, inf) and decreasing on (-inf, -1), (0,1).

To find concave up and down intervals we consult the second derivative, or d^2f/dx^2 = 24x^2-8. This has zeroes at 1/sqrt(3) and -1/sqrt(3). Concave up corresponds to positive values of the second derivative (-inf, -1/sqrt(3)), (1/sqrt(3), inf) while concave down corresponds to negative values of the second derivative (-1/sqrt(3), 1/sqrt(3)).

I feel as though I can't describe the sketch through text, but I hope this helps!

Answer 2

I have uploaded for you a FULL analysis of the function and it's graph in the resources section under the TOOLKIT

menu.

Their filenames are 2x^4 - 4x^2 + 1, and they are text files and jpeg picture file.

I also posted the link of your posting