F. Pleae help with a step by step guide. Thank you. Let f(x)=x^3 −12x + 5(a)  What is the y-intercept?(b)  Find all critical points for the first derivative.(c)  Use the second derivative test to determine whether these critical points are minima, maxima, or points of inflection.(d)  Use the second derivative to find any points of inflection.(e)  Sketch the graph, showing and labeling all extrema, points of inflection, and the y-intercept. You do not need to determine the x-intercepts.

Question
Asked Mar 24, 2019

F. Pleae help with a step by step guide. Thank you.

 

Let f(x)=x^3 −12x + 5

    1. (a)  What is the y-intercept?

    2. (b)  Find all critical points for the first derivative.

    3. (c)  Use the second derivative test to determine whether these critical points are minima, maxima, or points of inflection.

    4. (d)  Use the second derivative to find any points of inflection.

    5. (e)  Sketch the graph, showing and labeling all extrema, points of inflection, and the y-intercept. You do not need to determine the x-intercepts.

check_circle

Expert Answer

Step 1

Since we only answer up to 3 sub-parts, we’ll answer the first 3. Please resubmit the question and specify the other subparts (up to 3) you’d like answered

 

(a) y-intercept is the point where the graph crosses the y-axis i.e. when x=0

Plugging the value in the given function:

Hence, the y-intercept is 5

fullscreen
Step 2

(b) A point on a function is said to be a critical point if the value of the derivative at that point is 0

Applying derivative to the function,

A point on the given function is critical point if the value of the derivative is 0.

So,

Plugging the values in the function,

Therefore, the critical points are (2,-11) and (-2,21)

fullscreen
Step 3

(c)

As per the second derivative test, when a function has critical points at c and f(c)=0

If f ''(c)>0, then there is a local minimum at c

If f ''(c)<0, then there is a local maximum at c

If f ''(c)=0, then there is an inflection point at c

 

The second derivati...

fullscreen

Want to see the full answer?

See Solution

Check out a sample Q&A here.

Want to see this answer and more?

Solutions are written by subject experts who are available 24/7. Questions are typically answered within 1 hour.*

See Solution
*Response times may vary by subject and question.
Tagged in

Math

Calculus

Related Calculus Q&A

Find answers to questions asked by student like you

Show more Q&A add