Consider the function below. f(x) = ln(x4 + 27) (a) Find the interval of increase. (Enter your answer using interval notation.)
Find the interval of decrease. (Enter your answer using interval notation.) (b) Find the local minimum value(s). (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.) 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 inflection points. (x, y) = (smaller x-value) (x, y) = (larger x-value) Find the interval where the graph is concave upward. (Enter your answer using interval notation.) Find the intervals where the graph is concave downward. (Enter your answer using interval notation.)
a) The function is constantly increasing and is never decreasing
b) There is no local maximum or local minimum.
Step-by-step explanation:
To find the intervals of increasing and decreasing, we can start by finding the answers to part b, which is to find the local maximums and minimums. We do this by taking the derivatives of the equation.
f(x) = ln(x^4 + 27)
f'(x) = 1/(x^2 + 27)
Now we take the derivative and solve for zero to find the local max and mins.
f'(x) = 1/(x^2 + 27)
0 = 1/(x^2 + 27)
Since this function can never be equal to one, we know that there are no local maximums or minimums. This also lets us know that this function will constantly be increasing.
