解释:

Inflection points can only occur when the second derivative is zero or undefined. Here we have

.

Therefore possible inflection points occur atand. However, to have an inflection point we must check that the sign of the second derivative is different on each side of the point. Here we have

.

Hence, both are inflection points

解释:

Possible inflection points occur when. This occurs at three values,. However, to be an inflection point the sign ofmust be different on either side of the critical value. Hence, onlyare critical points.

解释:

A point of inflection is found where the graph (or image) of a function changes concavity. To find this algebraically, we want to find where the second derivative of the function changes sign, from negative to positive, or vice-versa. So, we find the second derivative of the given function

The first derivative using the power rule

is,

and the seconds derivative is

We then find where this second derivative equals.when.

We then look to see if the second derivative changes signs at this point. Both graphically and algebraically, we can see that the functiondoes indeed change sign at, and only at,, so this is our inflection point.

解释:

In order to find the points of inflection, we need to findusing the power rule,.

Now we set, and solve for.

To verify this is a true inflection point we need to plug in a value that is less than it and a value that is greater than it into the second derivative. If there is a sign change around the point than it is a true inflection point.

Let

Now let

自从negati改变从一个积极的迹象ve around the point, we can conclude it is an inflection point.

解释:

In order to find the points of inflection, we need to findusing the power rule.

Now to find the points of inflection, we need to set.

.

Now we can use the quadratic equation.

Recall that the quadratic equation is

,

where a,b,c refer to the coefficients of the equation.

In this case, a=12, b=0, c=-4.

Thus the possible points of infection are

.

现在我们检查或是否拐点need to plug in a value higher and lower than each point. If there is a sign change then the point is an inflection point.

To checklets plug in.

Thereforeis an inflection point.

Now lets checkwith.

Thereforeis also an inflection point.

解释:

In order to find the points of inflection, we need to findusing the power rule.

Now lets factor.

Now to find the points of inflection, we need to set.

.

From this equation, we already know one of the point of inflection,.

To figure out the rest of the points of inflection we can use the quadratic equation.

Recall that the quadratic equation is

, where a,b,c refer to the coefficients of the equation

.

In this case, a=20, b=0, c=-18.

Thus the other 2 points of infection are

To verify that they are all inflection points we need to plug in values higher and lower than each value and see if the sign changes.

Lets plug in

Since there is a sign change at each point, all are points of inflection.

解释:

In order to find the points of inflection, we need to find

Now we set.

.

This last statement says thatwill never be. Thus there are no points of inflection.

解释:

The points of inflection of a given function are the values at which thesecondderivative of the function are equal to zero.

The first derivative of the function is

, and the derivative of this function (the second derivative of the original function), is

.

Both derivatives were found using the power rule

.

Solving,.

To verify that this point is a true inflection point we need to plug in a value that is less than the point and one that is greater than the point into the second derivative. If there is a sign change between the two numbers than the point in question is an inflection point.

Lets plug in,

.

Now plug in

.

Therefore,is the only point of inflection of the function.

解释:

In order to find all the points of inflection, we first findusing the power rule twice,.

Now we set.

.

Now we factor the left hand side.

From this, we see that there is one point of inflection at.

For the point of inflection, lets solve for x for the equation inside the parentheses.

解释:

In order to find all the points of inflection, we first findusing the power rule twice.

Now we set.

Thus the points of inflection areand