Question

For the following function, a) give the coordinates of any critical points and classify each point as a relative maximum, a relative minimum, or neither; b) identify intervals where the function is increasing or decreasing; c) give the coordinates of any points of inflection; d) identify intervals where the function is concave up or concave down, and e) sketch the graph. $h(x) = 5x^3 - 15x$

          For the following function, a) give the coordinates of any critical points and classify each point as a
relative maximum, a relative minimum, or neither; b) identify intervals where the function is increasing
or decreasing; c) give the coordinates of any points of inflection; d) identify intervals where the function is
concave up or concave down, and e) sketch the graph.
$h(x) = 5x^3 - 15x$

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For the following function, a) give the coordinates of any critical points and classify each point as a
relative maximum, a relative minimum, or neither; b) identify intervals where the function is increasing
or decreasing; c) give the coordinates of any points of inflection; d) identify intervals where the function is
concave up or concave down, and e) sketch the graph.
h(x) = 5x^3 - 15x

Added by Ismael A.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Kathleen Carty

07/02/2022

Step 1

The derivative of h(x) is given by: h'(x) = 15x^2 - 15 Show more…

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It's one question with multiple parts. For the following function, a) give the coordinates of any critical points and classify each point as a relative maximum, a relative minimum, or neither; b) identify intervals where the function is increasing or decreasing; c) give the coordinates of any points of inflection; d) identify intervals where the function is concave up or concave down,and e sketch the graph. h()=5x3 - 15x

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00:03 We want to start by first finding the coordinates of any critical points and classifying those points as relative min or max or neither.

00:12 So we're going to take the derivative of this function, which gives me 15x squared minus 15.

00:20 We're going to set that function equal to zero.

00:25 I'm going to factor out 15.

00:30 That's going to give me x plus 1, x minus 1.

00:33 So my critical points are going to be at negative 1 and 1.

00:44 I can get the corresponding y coordinates by just plugging in those values.

00:49 If i plug in negative 1, i'm going to get out positive 10.

00:53 If i put in positive 1, i'm going to get out negative 10.

00:58 Now i want to be able to classify those as min or max points.

01:02 I plot them on a number line.

01:08 I'm going to choose values to the left and right, and i'm going to test those values in my first derivative.

01:16 So i want f prime of negative 2, f prime of 0, and f prime of positive 2.

01:22 I don't really care what number comes out.

01:25 I care whether it's positive or negative.

01:27 So f prime of negative 2 is going to give me 4 times 15, which is 60 minus 15, that's a positive.

01:36 I'm going to get the same result putting in positive 2.

01:40 If i put in 0, i get out negative 15, which is negative.

01:45 So that gives me these results.

01:47 Positive, negative, positive.

01:50 So that tells me my graph is increasing, decreasing, and then increasing.

01:56 So at negative 1 .10, i'm going to have a relative maximum.

02:07 At positive 1, negative 10, i'm going to have a relative maximum.

02:10 A relative minimum...

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