# 141.30 how many points of inflection will the graph of the function have

• MHB
• karush
In summary, the function given by the derivative $$f'(x)=\frac{1}{5}(x^2-4)^5-x^2$$ will have 5 points of inflection according to the second derivative $$f''(x)=2x((x^2-4)^4-1)$$, which can be factored to $$2 x (x^2 - 5) (x^2 - 3) (x^4 - 8 x^2 + 17) = 0$$. However, not all points where the second derivative is zero will be points of inflection, only those where the sign of the second derivative changes indicating a change in concavity. A point of inflection
karush
Gold Member
MHB
If the derivative of a function f is given by
$$f'(x)=\frac{1}{5}(x^2-4)^5-x^2$$
how many points of inflection will the graph of the function have?solution find $f"(x)$
$$f''(x)=2x((x^2-4)^4-1)$$
at $f''(x)=0$ we have factored
$$2 x (x^2 - 5) (x^2 - 3) (x^4 - 8 x^2 + 17) = 0$$
then
$$x=0\quad x=\pm\sqrt{3}\quad \pm\sqrt{5}$$
so we have 5 points of inflectionok I was wondering if this could be solved strictly by observation
also I used the $W\vert A$ to get $f"(x)$

the only thing I know about finding inflexions is they are zero points of the second direvative of a function
where concave <---> convex

Last edited:
A point of inflection will always be where the second derivative is zero, but not all places where the second derivative is zero will be a point of inflection. Any roots of the second derivative of even multiplicity will indicate that the second derivative does not change sign across that critical value. For the second derivative, a change in sign implies a change in the direction of concavity (up or down), and that's what is required.

For example, consider the function:

$$\displaystyle f(x)=x^4$$

We find:

$$\displaystyle f''(x)=12x^2$$

This has a root at $$x=0$$, but it is of multiplicity 2, and so we know the sign of $$f''$$ will not change across this critical value, therefore, the point $$(0,0)$$ is not a point of inflection.

thanks I didn't know that

however I assume that a point of inflection may exist even if it is a hole.

karush said:
thanks I didn't know that

however I assume that a point of inflection may exist even if it is a hole.

Yes, consider:

$$\displaystyle f(x)=\frac{x^4}{x}$$

The origin is still a point of inflection, even if the function is not defined there. :)

## 1. How do you determine the number of points of inflection on a graph?

To determine the number of points of inflection on a graph, you must first find the second derivative of the function. Then, set the second derivative equal to zero and solve for the values of x. The number of distinct solutions will be the number of points of inflection on the graph.

## 2. What is a point of inflection on a graph?

A point of inflection is a point on a graph where the concavity changes. This means that the graph changes from being concave up to concave down, or vice versa. At a point of inflection, the tangent line is horizontal and the second derivative of the function is equal to zero.

## 3. Can a graph have more than one point of inflection?

Yes, a graph can have multiple points of inflection. This occurs when the concavity changes multiple times on the graph. The number of points of inflection can be determined by finding the number of distinct solutions to the second derivative of the function.

## 4. What does the second derivative of a function represent?

The second derivative of a function represents the rate of change of the slope of the original function. It can also be thought of as the rate of change of the concavity of the graph. A positive second derivative indicates a concave up graph, while a negative second derivative indicates a concave down graph.

## 5. How does the number of points of inflection affect the graph of a function?

The number of points of inflection can affect the shape and behavior of a graph. If there are no points of inflection, the graph will either be always concave up or always concave down. If there is one point of inflection, the graph will change concavity at that point. If there are multiple points of inflection, the graph may have several changes in concavity.

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