Find the absolute maxima and minima for f(x) on the interval [a, b].

Question

Find the absolute maxima and minima for f(x) on the interval [a, b]

f (x) = x³ + x² − x + 8, [-2,0]

absolute maximum    (x,y) =

absolute minimum    (x,y) =

Answer 1

Finding the absolute maximum is looking for the local maximum and testing the endpoints. To find the local maximum is akin to finding the derivative and setting it to zero. We use the second derivative as a test for max/min

f(x) = x³ + x² + x - 8 for the interval [-2, 0]

f'(x) = 3x² + 2x + 1 = 0 --> x = [ -2 +/- sqrt(4 - 12) ] /2 = -1 +/ sqrt(-2) = -1 +/- 2i. (complex numbers)

Since the solutions of the local max/min are complex, there are no local max/min values

Now testing the endpoints...

f(-2) = -8 + 4 - 2 - 8 = -14

f(0) = 0 + 0 + 0 - 8 = -8

Absolute maximum is at (0,-8)