# Stationary Point - Possible Values

• BAH0003
In summary, a stationary point is a point on a graph where the gradient is equal to zero, indicating no increase or decrease. To find stationary points, the derivative must be set to zero and solved for the variable. The possible values of a stationary point include the x and y coordinates, as well as the type (maximum, minimum, or point of inflection). The type can be determined by looking at the graph or using the second derivative test. Stationary points are important because they provide valuable information about the behavior of a function and can be used to optimize functions in various fields.
BAH0003
Homework Statement
Hi,

I am having trouble with part d) of this question. It follows on with other parts of a question which I have attached. I have written that 'p' can indeed have stationary points but am not sure what the possible values of 'p' could be. If anyone can list these possible values that would be great, as this is apart of a big assignment that is due soon. Please note this is from question 4 from the first picture.

Thank You and Kind Regards.
Relevant Equations
Please refer to the sheet attatched
Please refer to the image attached

BAH0003 said:
I have written that 'p' can indeed have stationary points but am not sure what the possible values of 'p' could be.
'p' does not have stationary points. It is the number of stationary points that the quartic has.
Think about the derivative of the quartic.

sysprog

## What is a stationary point?

A stationary point is a point on a graph where the slope (or gradient) of the curve is equal to zero. This means that the curve is neither increasing nor decreasing at that point.

## What are possible values at a stationary point?

The possible values at a stationary point depend on the type of stationary point. For a maximum stationary point, the possible value is the highest point on the curve. For a minimum stationary point, the possible value is the lowest point on the curve. For a point of inflection, the possible value is the point where the curve changes from being concave up to concave down, or vice versa.

## How do you find the possible values at a stationary point?

To find the possible values at a stationary point, you can use the first or second derivative test. The first derivative test involves finding the derivative of the function and setting it equal to zero. The solutions to this equation will give the x-values of the stationary points. The second derivative test involves finding the second derivative of the function and evaluating it at the x-values found in the first derivative test. If the second derivative is positive, the stationary point is a minimum, and if it is negative, the stationary point is a maximum.

## Why are stationary points important in mathematics?

Stationary points are important in mathematics because they can help us find critical information about a function, such as maximum and minimum values, points of inflection, and intervals of increase or decrease. They are also used in optimization problems, where we want to find the maximum or minimum value of a function within a given range.

## Can a function have more than one stationary point?

Yes, a function can have more than one stationary point. In fact, it is common for functions to have multiple stationary points, especially in more complex functions. These points can be classified as maximum, minimum, or points of inflection. It is important to identify all stationary points in a function in order to fully understand its behavior.

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