Let 𝑓(𝑥)=𝑥2+2𝑥.Determine the absolute maximum and absolute minimum values of 𝑓 on the interval [12,2].

To find the absolute maximum and minimum values of a function on a closed interval, we need to evaluate the function at its critical points and endpoints, and compare the function values.

Step 1: Find the derivative of the function. f'(x) = 2x + 2

Step 2: Set the derivative equal to zero and solve for x to find critical points. 2x + 2 = 0 2x = -2 x = -1

However, x = -1 is not in the interval [1/2, 2].

Step 3: Evaluate the function at the endpoints and at the critical points within the interval. f(1/2) = (1/2)^2 + 2*(1/2) = 1/4 + 1 = 1.25 f(2) = 2^2 + 2*2 = 4 + 4 = 8

Step 4: Compare the function values. The absolute maximum of f on the interval [1/2, 2] is 8 and the absolute minimum is 1.25.