Find the absolute maximum and absolute minimum values of f on the given interval. f(x) = 6x3 - 9x² – 216x + 4, [-4, 5)

Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Find the absolute maximum and absolute minimum values of f on the given interval.
f(x) = 6x3 – 9x² – 216x + 4,
[-4, 5]
Step 1
The absolute maximum and minimum values of f occur either at a critical point inside the interval or at an endpoint of the interval. Recall that a critical point is a point where f '(x) = 0 or is
undefined. We begin by finding the derivative of f.
f '(x)
18x – 18x – 216
18:2 – 18x
216
Step 2
We now solve f '(x) = 0 for x, which gives the following critical numbers. (Enter your answers as a comma-separated list.)
X =
-3,4
-3, 4
Step 3
We must now find the function values at the critical numbers we just found and at the endpoints of the interval [-4, 5].
f(-3)
= 409
409
f(4) =
-620
-620
f(-4)
= 340
340
f(5) = -551
-551
Step 4
Therefore, on the interval [-4, 5], the absolute minimum of f(x) is
and the absolute maximum is
Transcribed Image Text:Find the absolute maximum and absolute minimum values of f on the given interval. f(x) = 6x3 – 9x² – 216x + 4, [-4, 5] Step 1 The absolute maximum and minimum values of f occur either at a critical point inside the interval or at an endpoint of the interval. Recall that a critical point is a point where f '(x) = 0 or is undefined. We begin by finding the derivative of f. f '(x) 18x – 18x – 216 18:2 – 18x 216 Step 2 We now solve f '(x) = 0 for x, which gives the following critical numbers. (Enter your answers as a comma-separated list.) X = -3,4 -3, 4 Step 3 We must now find the function values at the critical numbers we just found and at the endpoints of the interval [-4, 5]. f(-3) = 409 409 f(4) = -620 -620 f(-4) = 340 340 f(5) = -551 -551 Step 4 Therefore, on the interval [-4, 5], the absolute minimum of f(x) is and the absolute maximum is
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