f(x) = 1 + 2x^2 − x^4

Question

Consider the function below.

f(x) = 1 + 2x² − x⁴

(a) Find the local minimum value(s). (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)

(b) Find the local maximum value(s). (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)

(c) Find the interval where the graph is concave upward. (Enter your answer using interval notation.)

Find the interval where the graph is concave downward. (Enter your answer using interval notation.)

Answer 1

For f(x) equal to 1 + 2x² − x⁴, Establish:

First Derivative f'(x) is 4x − 4x³;

Second Derivative f''(x) is 4 − 12x².

4x − 4x³ or 4x(1 − x²) goes to 0 at x = 0, x = -1, & x = 1.

For x = 0, 4 − 12x² is 4, greater than 0, which indicates a minimum for f(x) at x = 0.

For x = -1, 4 − 12x² is -8, less than 0, which indicates a maximum for f(x) at x = -1.

For x = 1, 4 − 12x² is also -8, less than 0, which indicates a maximum for f(x) at x = 1.

A graph of f(x) = 1 + 2x² − x⁴ shows the function crossing the x-axis around x = 1.5 and x = -1.5.

Isaac Newton's Method gives these "zeroes" to extreme accuracy as:

x = 1.553773974;

x = -1.553773974.

Intervals of concavity are:

concave downward from -1.553773974 < x < -1;

concave upward from -1 < x < 0;

concave upward from 0 < x < 1;

concave downward from 1 < x < 1.553773974.