Use a graphing utility to graph the function and find the absolute extrema of the function on the given interval.

Question

Round your answers to three decimal places. If an answer does not exist, enter DNE.)

f(x) = x⁴ − 2x³ + x + 1,    [−1, 3]

minima ? (smaller x-value)

? (larger x-value)

maximum?

Answer 1

Absolute extrema for a function on a continuous closed interval happen at either the endpoints of the interval or at critical numbers in the interval (x-values where the tangent line is horizontal--slope of zero).

To find the critical numbers set f'(x) = 0 and solve for x.

f'(x) = 4x³ - 6x² + 1

4x³ - 6x² + 1 = 0

This is not so easy to solve for x (looking at a graph or using Newton's method reveals):

x₁ ≈ -0.366

x₂ = 0.5

x₃ ≈ 1.366

If you instead first use the rational root theorem, you might try synthetic division with a list of possible rational roots: {±1,±1/2,±1/4} and discover that 1/2 is indeed a root. The synthetic division shows that the remaining trinomial factor is: (4x² - 4x - 2). Setting that equal to zero and solving for x (quadratic formula?) shows the exact values as [1 ± √3]/2.

The list of x values to test for absolute extrema on [-1, 3] are:

-1, -0.366, 1/2, 1.366, 3.

This graph reveals that the absolute mins occur at the irrational x-values, and the absolute max at x = 3.

desmos.com/calculator/9cn7t4w8ho

Visit the graph to see what those min and max values are.