{Use of Tech} Absolute maxima and minima


a. Find the critical points of f on the given interval.
b. Determine the absolute extreme values of f on the given interval.
c. Use a graphing utility to confirm your conclusions.


f(x) = 2ᶻ sin x on [-2,6]

Verified step by step guidance

1. To find the critical points of the function f(x) = 2^x sin(x) on the interval [-2, 6], first compute the derivative f'(x). Use the product rule: if u(x) = 2^x and v(x) = sin(x), then f'(x) = u'(x)v(x) + u(x)v'(x).
2. Calculate the derivatives: u'(x) = 2^x ln(2) and v'(x) = cos(x). Substitute these into the product rule to get f'(x) = 2^x ln(2) sin(x) + 2^x cos(x).
3. Set the derivative f'(x) to zero to find the critical points: 2^x ln(2) sin(x) + 2^x cos(x) = 0. Factor out 2^x to simplify: 2^x (ln(2) sin(x) + cos(x)) = 0. Since 2^x is never zero, solve ln(2) sin(x) + cos(x) = 0 for x in the interval [-2, 6].
4. Evaluate the function f(x) at the critical points found in the previous step, as well as at the endpoints of the interval, x = -2 and x = 6. This will help determine the absolute maximum and minimum values of f(x) on the interval.
5. Use a graphing utility to plot f(x) = 2^x sin(x) over the interval [-2, 6]. Verify that the critical points and endpoints correspond to the absolute maximum and minimum values found in the previous step.