Question

Consider the function f(x) = 5x² - 8x +7, 0≤ x ≤ 8.

The absolute maximum of f(x) (on the given interval) is at x =
and the absolute minimum of f(x) (on the given interval) is at x =

Answer 1

• maximum: x = 8
• minimum: x = 0.8

Explanation:

You want the x-coordinates of the absolute extrema of the function f(x) = 5x² -8x +7 on the interval [0, 8].

Absolute extrema

The absolute extrema of a function on an interval will lie at the ends of the interval or at a turning point within the interval.

For a quadratic ax²+bx+c, the turning point is at x=-b/(2a). For the given function, the turning point is at ...

x = -(-8)/(2·5) = 0.8

This lies within the interval, and represents the location of the absolute minimum of this function whose graph opens upward.

Interval ends

The graph of the function is symmetrical about the vertical line through the turning point. The function increases as x-values are farther from the turning point, so the end of the interval farthest from x = 0.8 will be the location of the absolute maximum. That is at x = 8.

The absolute minimum at x = 0.8; the absolute maximum is at x = 8.