{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 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, to determine the absolute maximum and minimum values.
5. Use a graphing utility to plot f(x) = 2^x sin(x) over the interval [-2, 6] and visually confirm the locations and values of the absolute maxima and minima identified in the previous steps.

Key Concepts

Critical Points

Critical points of a function occur where its derivative is zero or undefined. These points are essential for identifying local maxima and minima, as they represent potential locations where the function's behavior changes. To find critical points, one must first compute the derivative of the function and solve for the values of x that satisfy the condition.

Absolute Extrema

Absolute extrema refer to the highest and lowest values of a function on a given interval. To determine these values, one must evaluate the function at its critical points and at the endpoints of the interval. The largest value found will be the absolute maximum, while the smallest will be the absolute minimum, providing a complete picture of the function's behavior over the specified range.

Graphing Utility

A graphing utility is a software tool or calculator that allows users to visualize mathematical functions and their properties. By plotting the function, one can easily observe its critical points, local maxima, and minima, as well as confirm analytical findings. This visual representation aids in understanding the function's overall behavior and verifying calculations made during the analysis.