Question

Consider the equation below. f(x) = 9 sin(x) + 9 cos(x), 0 ≤ x ≤ 2π (a) Find the interval on which f is increasing. (Enter your answer using interval notation.) Find the interval on which f is decreasing. (Enter your answer using interval notation.) Correct: Your answer is correct. (b) Find the local minimum and maximum values of f. local minimum value Correct: Your answer is correct. local maximum value Correct: Your answer is correct. (c) Find the inflection points. (x, y) = (smaller x-value) (x, y) = (larger x-value) Find the interval on which f is concave up. (Enter your answer using interval notation.) Correct: Your answer is correct. Find the interval on which f is concave down. (Enter your answer using interval notation.

Answer 1

To determine where the function f(x) = 9 sin(x) + 9 cos(x) is increasing or decreasing, we analyze its derivative. Local minima and maxima are identified where the derivative changes sign. Second derivatives reveal inflection points and intervals of concavity.

Step-by-step explanation:

To find the intervals on which the function f(x) = 9 sin(x) + 9 cos(x) is increasing or decreasing, we first need to find its derivative f'(x). The derivative will tell us the slope of the function at any given point. Where the derivative is positive, the function is increasing, and where it is negative, the function is decreasing.

To find the local minimum and maximum values, we look for points where f'(x) changes sign, which often corresponds to the derivative being zero or undefined. These points are where the function switches from increasing to decreasing (local maxima) or from decreasing to increasing (local minima).

To find inflection points and intervals of concavity, we take the second derivative f''(x). Where f''(x) is positive, the function is concave up, and where f''(x) is negative, the function is concave down. Inflection points are where f''(x) changes sign.