Question

For a given function, determine where it is increasing and decreasing:

a) At critical points
b) In the interval (−[infinity], [infinity])
c) At local maxima and minima
d) At points of inflection

Answer 1

To determine where a function is increasing and decreasing, you need to consider critical points, intervals, local maxima and minima, and points of inflection.

Step-by-step explanation:

a) At critical points: Find the points where the derivative of the function equals zero or is undefined. These are the potential critical points. Evaluate the derivative on either side of each potential critical point to determine if the function is increasing or decreasing.

b) In the interval (-∞, ∞): Evaluate the derivative on each interval to determine if the function is increasing or decreasing.

c) At local maxima and minima: Find the points where the derivative changes sign from positive to negative or from negative to positive. These are the potential local maxima and minima. Evaluate the derivative on either side of each potential maxima or minima to determine if the function is increasing or decreasing.

d) At points of inflection: Find the points where the second derivative of the function equals zero or is undefined. These are the potential points of inflection. Evaluate the second derivative on either side of each potential point of inflection to determine if the function is concave up or concave down.