- #1
angela107
- 35
- 2
- Homework Statement
- Will my local minimum be x=1?
- Relevant Equations
- n/a
How do you find a, b, and show that that đť‘“(đť‘Ą) has a local min value?
In summary, the conversation discusses the solution to the question "the graph of the function ##f(x)=ax^2+\frac{b}{x}## has a horizontal tangent at point (1,3). Find ##a##, ##b##, and show that ##f(x)## has a local minimum value at ##x=1##." The individual providing the solution also asks for confirmation on their answer and for the question to be provided in the future.
- #2
BvU
Science Advisor
Homework Helper
- 15,986
- 4,806
You get a whole story but you don't believe the happy end ? What is your real question / doubt ?angela107 said:
- #3
RPinPA
Science Advisor
Homework Helper
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Is the attached picture your solution? You didn't tell us the question so it takes some guesswork to figure out what the question was. The attached text says "Since there is a horizontal tangent at the point (1,3)". Was that part of the question, some information you were given? And I have to assume that the form of f(x) you gave us was what you were given.
Given those two things, your solution for a and b is correct, and your conclusion about x = 1 being a local minimum is also correct.
Was that what you were asked to prove? If so, you have done so.
Could you in future please provide the question?
Given those two things, your solution for a and b is correct, and your conclusion about x = 1 being a local minimum is also correct.
Was that what you were asked to prove? If so, you have done so.
Could you in future please provide the question?
- #4
angela107
- 35
- 2
sorry, the question is "the graph of the function ##f(x)=ax^2+\frac{b}{x}## has a horizontal tangent at point (1,3). Find ##a##, ##b##, and show that ##f(x)## has a local minimum value at ##x=1##.RPinPA said:
- #5
angela107
- 35
- 2
I've checked my work, but can you double-check?BvU said:
1. What is the process for finding the values of a and b in order to show that đť‘“(đť‘Ą) has a local min value?
The process for finding the values of a and b involves taking the derivative of đť‘“(đť‘Ą) and setting it equal to 0. This will give you the critical points of the function. Then, you can use the second derivative test to determine if the critical point is a local minimum. If the second derivative is positive, the critical point is a local minimum and the values of a and b can be found.
2. How do you know if đť‘“(đť‘Ą) has a local min value?
If the second derivative of đť‘“(đť‘Ą) is positive at a critical point, it means that the function is concave up and the critical point is a local minimum. This means that the function has a local min value at that point.
3. Can a function have more than one local min value?
Yes, a function can have multiple local min values. This occurs when the function has multiple critical points where the second derivative is positive. Each of these critical points will be a local minimum for the function.
4. Is it necessary to find the values of a and b in order to show that đť‘“(đť‘Ą) has a local min value?
No, it is not necessary to find the values of a and b to show that đť‘“(đť‘Ą) has a local min value. The second derivative test can be used to determine if a critical point is a local minimum without explicitly finding the values of a and b.
5. Can a function have a local min value at a point where the second derivative is 0?
Yes, a function can have a local min value at a point where the second derivative is 0. This occurs when the function is flat at that point, meaning it has a horizontal tangent. In this case, the critical point is still a local minimum, but the second derivative test cannot be used to determine this.
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