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3) Consider the function \begin{equation} g(x) = \begin{cases} |x+1| & \text{if } x \le 1\\ (x-2)^2 + 1 & \text{if } x > 1 \end{cases} \end{equation} a) (2 points) Sketch the graph of the continuous function b) (3 points) Sketch the graph of $g'(x)$. Use open circles (o) as needed c) (1 point) What are the critical points of $g(x)$? d) (1 point) On what internals is $g(x)$ increasing? e) (1 point) On what internals is $g(x)$ decreasing? f) (1 point) On what intervals is $g(x)$ concave up? g) (1 point) On what intervals is $g(x)$ concave down?

          3) Consider the function
\begin{equation*} g(x) = \begin{cases} |x+1| & \text{if } x \le 1\\ (x-2)^2 + 1 & \text{if } x > 1 \end{cases} \end{equation*}
a) (2 points) Sketch the graph of the continuous function
b) (3 points) Sketch the graph of $g'(x)$. Use open circles (o) as needed
c) (1 point) What are the critical points of $g(x)$?
d) (1 point) On what internals is $g(x)$ increasing?
e) (1 point) On what internals is $g(x)$ decreasing?
f) (1 point) On what intervals is $g(x)$ concave up?
g) (1 point) On what intervals is $g(x)$ concave down?

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3) Consider the function

g(x) = |x+1| if x ≤ 1
(x-2)^2 + 1 if x > 1

a) (2 points) Sketch the graph of the continuous function
b) (3 points) Sketch the graph of g'(x). Use open circles (o) as needed
c) (1 point) What are the critical points of g(x)?
d) (1 point) On what internals is g(x) increasing?
e) (1 point) On what internals is g(x) decreasing?
f) (1 point) On what intervals is g(x) concave up?
g) (1 point) On what intervals is g(x) concave down?

Added by John F.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Vincenzo Zaccaro

07/03/2023

Step 1

$$g'(x) = \begin{cases} -1 & \text{if } x < -1 \\ 1 & \text{if } -1 < x \le 1 \\ 2(x-2) & \text{if } x > 1 \end{cases}$$ Show more…

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3) Consider the function $$g(x) = \begin{cases} |x + 1| & \text{if } x \le 1 \\ (x - 2)^2 + 1 & \text{if } x > 1 \end{cases}$$ a) (2 points) Sketch the graph of the continuous function b) (3 points) Sketch the graph of g'(x). Use open circles (o) as needed c) (1 point) What are the critical points of g(x)? d) (1 point) On what internals is g(x) increasing? e) (1 point) On what internals is g(x) decreasing? f) (1 point) On what intervals is g(x) concave up? g) (1 point) On what intervals is g(x) concave down?

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Transcript

00:01 In this exercise we have our function f of x equal to 2x cubed minus 3x squared minus 12x.

00:12 Let's compute the derivative of this function.

00:17 The derivative is 6x squared minus 6x minus 12.

00:24 This one can be written as 6 multiplied by x squared minus x minus 2.

00:35 Perfect.

00:36 Now we need to find the zeros.

00:40 Ok, f prime of x is equal to 0 if and only if, thanks to the quadratic formula, x is equal to 1 plus or minus square root of 9 over 2.

00:57 So we get 1 plus 3 over 2 which is 2, 1 minus 3 over 2 which is negative 1.

01:07 Perfect.

01:08 Ok, at this point what do we know? we know that this is going to be the sign chart for f prime.

01:18 Well, we know that f prime is 0 at negative 1 and 2.

01:24 Negative here and positive here.

01:28 Perfect.

01:30 So this thing shows that the local extreme of f are x equal to negative 1 and 2.

01:49 Moreover, we know that f is increasing where f prime is positive.

01:55 So, increasing on the union of the intervals negative infinity, negative 1 and 2 comma infinity.

02:06 And clearly f is decreasing on the complement.

02:12 So, on negative 1 comma 2...

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