Final answer:
The absolute maximum value of the function f(x) = 4 - 7x² is 4, occurring at x = 0, and the absolute minimum value is -171, occurring at x = -5.
Step-by-step explanation:
To find the absolute maximum and absolute minimum values of the function f(x) = 4 - 7x², for −5 ≤ x ≤ 2, we first recognize that this is a downward-opening parabola, since the coefficient of the x² term is negative. To find the extrema, we need to evaluate the function at the endpoints and at any critical points within the interval.
Since the function is a parabola, it has a vertex, which represents either a maximum or minimum. For the function f(x) = 4 - 7x², the vertex occurs at x = 0 (since there is no x-term, and thus the axis of symmetry is x = 0). At this point, the value of the function is f(0) = 4. This is a likely candidate for the maximum, because the function opens downward.
Next, we check the value of the function at the end points:
- f(-5) = 4 - 7(-5)² = 4 - 175 = -171
- f(2) = 4 - 7(2)² = 4 - 28 = -24
Comparing the values, we see that the absolute maximum value of the function on the interval is indeed 4, and it occurs at x = 0. The absolute minimum value is -171, and this occurs at x = -5.