# 12.8.3. Curve Sketching

Find the values of and that makes the graph of the function has an inflection point at the point .

• A,
• B,
• C,
• D,

### Example

Find the values of and that makes the graph of the function has an inflection point at the point .

### Solution

If is a function for which over an interval, then the graph of is convex upwards over this interval. Likewise, if is a function for which over an interval, then the graph of is convex downwards over this interval. A point of inflection on a curve is a point where the curve changes from convex downwards to convex upwards, or vice versa, so at a point of inflection, either , or is undefined. First, use implicit differentiation to find the first derivative for the function as follows:

Now use implicit differentiation to find the second derivative of the function .

Next, since the graph of the function has an inflection point at , substitute 0 for , 3 for , and for in the equation and simplify.

Now substitute 3 for and for in the equation and simplify.

Next, to solve the system of equations , multiply the bottom equation by 2 to get the system , add the two equations to get , and divide both sides by 18 to get . Finally, substitute 8 for in the equation and solve for as follows:

Thus, the values of and that make the graph of the function have an inflection point at are 8 and 3, respectively.

0
correct
0
incorrect
0
skipped