A function f is defined on the interval [0,4], and its derivative is e^sinx-2cos3x
a. on what interval is f increasing?
b. at what value(s) of x does f have a local maxima?
c. how many points of inflection does f have?
*calculator is allowed for this problem
To determine the answers to these questions, we will need to find the antiderivative of the given derivative function. The antiderivative of e^sin(x) is -e^sin(x), and the antiderivative of -2cos(3x) is -(2/3)sin(3x).
a. To find on what interval f is increasing, we need to check where the derivative is positive. Therefore, we need to find the critical points by solving the equation: e^sin(x) - 2cos(3x) = 0.
Since this equation is not easy to solve analytically, we can utilize a numerical method or a calculator. You can use a graphing calculator or an online graphing tool to plot the graph of the derivative function and observe the intervals where it is above the x-axis. Those intervals will correspond to the intervals on which f is increasing.
b. To find the local maxima of f, we need to find the critical points where the derivative changes sign. We can use the same technique as in part 'a' and solve the equation e^sin(x) - 2cos(3x) = 0. Once we have the critical points, we can check the sign of the derivative on each interval bounded by the critical points to determine if there is a local maximum.
c. To find the points of inflection, we need to find where the concavity of the function changes. This occurs when the second derivative changes sign. To obtain the second derivative, we differentiate the given derivative function e^sin(x) - 2cos(3x). The result will be a new function that represents the second derivative. We can then identify the critical points for this function and analyze the concavity between those critical points.
Again, we can use a graphing calculator or an online graphing tool to plot the second derivative and observe the intervals where it changes sign. The number of sign changes will indicate the number of points of inflection.
a. It increase at that 3pt line where I get buckets
b. all max no brakes
c. I had 81 point boi