# Given ##f(x)=3x^5-5x^3##, find all critical points.

• angela107
In summary, critical points are points on a graph where the derivative of a function is zero or undefined. They are important because they can indicate the function's maximum, minimum, or inflection points. To find them, you must take the derivative of the function and set it equal to zero. Critical points can help identify key features of a function and can also be classified using the first or second derivative test. A function can have multiple critical points, and the second derivative test can determine if a critical point is a maximum, minimum, or inflection point.
angela107
Homework Statement
Given ##f(x)=3x^5-5x^3##, find all critical points and identify any local
max/min point.
Relevant Equations
n/a
Is my work right?

Check your factorisation of y'' (but you do not seem to have used it).
Other than that, looks fine.

Last edited:
I think it's fine, but I agree with @haruspex that you have to check the factorisation of ##y''##.
Just a friendly reminder: there are critical points as well where the derivative doesn't exist. In this case, the first derivative exists for every ##x##, so you don't have to matter.

## 1. What is a critical point?

A critical point is a point on a graph where the slope (or derivative) of the function is equal to zero, or where the slope is undefined. It can also be described as a point where the function changes from increasing to decreasing, or vice versa.

## 2. How do you find critical points?

To find critical points, we need to take the derivative of the given function and set it equal to zero. Then, we solve for the variable to find the x-values of the critical points. We also need to check for any points where the derivative is undefined, as these can also be critical points.

## 3. What is the significance of critical points?

Critical points are important because they can give us information about the behavior of a function. For example, a critical point can indicate where a function reaches a maximum or minimum value, or where the function changes direction.

## 4. Can a function have more than one critical point?

Yes, a function can have multiple critical points. This can happen when a function has multiple peaks or valleys, or when the function has a point of inflection where the slope changes from positive to negative or vice versa.

## 5. How do you determine if a critical point is a maximum or minimum?

To determine if a critical point is a maximum or minimum, we can use the second derivative test. If the second derivative is positive at a critical point, then the point is a minimum. If the second derivative is negative, then the point is a maximum. If the second derivative is zero, then the test is inconclusive and we need to use other methods, such as checking the behavior of the function near the critical point.

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