The inflection point of the curve y = x4 − 8x3 + 24x2 + 7x − 3 is?

So I got (1,21)

I was wondering if this is correct because sometimes I will be doing something completely wrong and not even realize it.

To find the inflection point of a curve, you need to locate the x-coordinate(s) at which the concavity of the curve changes.

In order to determine the points of inflection, follow these steps:

Step 1: Find the second derivative of the function. The second derivative will tell us about the concavity of the curve.
Step 2: Set the second derivative equal to zero and solve for x to find possible inflection points.
Step 3: Check the concavity on both sides of the possible inflection point(s) to confirm whether they are indeed inflection points.
Step 4: If the concavity changes at the given point, it is an inflection point. If not, repeat steps 2 and 3.

Let's apply these steps to your function, y = x^4 − 8x^3 + 24x^2 + 7x − 3:

Step 1: Find the second derivative

The first derivative of y = x^4 − 8x^3 + 24x^2 + 7x − 3 is:
y' = 4x^3 - 24x^2 + 48x + 7

Now, take the derivative of the first derivative to get the second derivative:
y'' = 12x^2 - 48x + 48

Step 2: Set the second derivative equal to zero and solve for x

12x^2 - 48x + 48 = 0

Step 3: Check the concavity on both sides of the possible inflection point(s)

To determine concavity, take a test point on both sides of the possible inflection point(s). These test points should give you positive values on one side and negative on the other. Substitute the test point values into the second derivative equation, y'', and check if the result changes sign.

Step 4: Determine if it is an inflection point

If the concavity changes sign at the point(s) found in step 2, then they are indeed inflection points.

By solving the equation from step 2, you can find the possible x-coordinate(s) of inflection point(s). It seems you have found one point, which is (1, 21).

To confirm if it is a true inflection point, you need to check the concavity on both sides of x = 1 using test points. Calculate the value of y'' for values of x greater than and less than 1, and determine if the sign changes. If the concavity changes sign at x = 1, then (1, 21) is correct inflection point for the given curve.

Hope this helps! Let me know if you have further questions.