Domain: ( \infty, \infty) Derivative =0 at x=1 Derivative undefined at x= -3 and x=4 Test...

Question:

Domain:{eq}(-\infty, \infty) {/eq}

Derivative {eq}=0 {/eq} at {eq}x=1 {/eq}

Derivative undefined at {eq}x=-3 {/eq} and {eq}x=4 {/eq}

Test numbers:

{eq}f'(0)=-12 \\ f'(4)=28 \\ f'(10)=20 \\ f'(1.25)=-5,4321 {/eq}

a. What are the critical numbers if any?

b. Where is the function increasing?

c. Where is the function decreasing?

d. Where is the relative maximum?

e. Where is the relative minimum?

Behavior of Function on Domain


Given function values and function derivative values at different points on its domain we determine the function's critical points. Then using the critical points and earlier information we estimate the intervals over which the function is increasing, decreasing and constant. The concepts used from Calculus include that a function is increasing over an interval on which its derivative is positive and similarly decreasing and constant over an interval on which its derivative is negative and zero, respectively. Finally, using the function information we determine the relative maximum and the relative minimum of the function. The knowledge of the function that is gained from this question can help us sketch a graph of this function as well.


Answer and Explanation:


a. The critical numbers are where the derivative of the function is undefined or zero. These will be the points:

{eq}x=-3, \; 1, \; 4. {/eq}


b. The function is increasing where its derivative is positive for certain. This would be the following interval:

{eq}[4,10]. {/eq}


c. The function is decreasing where its derivative is negative for certain. This would be the following interval:

{eq}[0,1.25]. {/eq}


d. Relative Maximum is UNKNOWN.


e. Relative Minimum appears to occur at x = 1 since the second derivative at this point is positive.


Learn more about this topic:

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Finding Critical Points in Calculus: Function & Graph

from CAHSEE Math Exam: Tutoring Solution

Chapter 8 / Lesson 9
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