1st and 2nd derivative of a cubic function. and graphing

• chubbyorphan
In summary, the conversation discusses a graph of the function h(x) and asks for information on its intervals of increase and decrease, local maximum and minimum points, concavity, and inflection points. The attempt at a solution provides the following information: a) h(x) is not decreasing and is increasing at x < 2 and x > 2, b) there are no maximum or minimum points, c) h(x) is concave up at x > 2 and concave down at x < 2, and d) the inflection point occurs at x = 2. The conversation also includes sketches of h’(x) and h’’(x) for verification.
chubbyorphan
Hey forum, I know this is an easy one, but it's been a while for me :P

Homework Statement

Given the following graph of h(x)

a) The intervals where h(x) is increasing and decreasing
b) The local maximum and minimum points of h(x)
c) The intervals where h(x) is concave up and concave down
d) The inflection points of h(x)
e) Sketch the graphs of h’(x) and h’’(x)

The Attempt at a Solution

So far I have:

a)The function is not decreasing, and hence h’(x) is not < 0
The function is increasing, and hence h’(x) > 0 when x < 2 and x > 2

b) there are no maximum or minimum points

c)The function is concave up and hence h’’(x) > 0 when x > 2
The function is concave down and hence h’’(x) < 0 when x < 2

d)The inflection point occurs at x = 2

If someone could check this for me I would really appreciate it..
its especially part b) that I'm worried about
I know this question isn't very hard but it's been a long time since I've worked with graphs and my confidence is lacking. Thanks to anyone who can share some insight!

here are sketches for my graphs:

how do they look?
thanks again!

1. What is the formula for finding the first derivative of a cubic function?

The formula for finding the first derivative of a cubic function is f'(x) = 3ax^2 + 2bx + c, where a, b, and c are the coefficients of the cubic function.

2. How do you interpret the first derivative of a cubic function graphically?

The first derivative of a cubic function represents the slope of the tangent line at any given point on the graph of the function. This means that the first derivative can tell us the rate of change of the function at a specific point.

3. What is the process for finding the second derivative of a cubic function?

To find the second derivative of a cubic function, first find the first derivative using the formula f'(x) = 3ax^2 + 2bx + c. Then, take the derivative of the first derivative, which will result in f''(x) = 6ax + 2b. This is the second derivative of the cubic function.

4. How does the second derivative help in graphing a cubic function?

The second derivative can help us determine the concavity of the graph of a cubic function. If the second derivative is positive, the graph is concave up, and if the second derivative is negative, the graph is concave down. This information can help us accurately sketch the graph of the cubic function.

5. Can a cubic function have more than one point of inflection?

Yes, a cubic function can have more than one point of inflection. This occurs when the second derivative changes sign more than once, causing the concavity of the graph to change more than once. In general, a polynomial function of degree n can have up to n-2 points of inflection.

• Calculus and Beyond Homework Help
Replies
1
Views
611
• Calculus and Beyond Homework Help
Replies
4
Views
1K
• Calculus and Beyond Homework Help
Replies
8
Views
751
• Calculus and Beyond Homework Help
Replies
5
Views
489
• Calculus and Beyond Homework Help
Replies
6
Views
1K
• Calculus and Beyond Homework Help
Replies
4
Views
312
• Calculus and Beyond Homework Help
Replies
6
Views
733
• Calculus and Beyond Homework Help
Replies
22
Views
2K
• Calculus and Beyond Homework Help
Replies
15
Views
2K
• Calculus and Beyond Homework Help
Replies
2
Views
1K