# 12.8.3. Curve Sketching

Find the point of inflection of the function .

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### Example

Find the point of inflection of the function .

### 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. First, find for the function when , keeping in mind that when , the function can be rewritten as .

Now find for the function when .

Next, find for the function when , keeping in mind that when , the function can be rewritten as , or .

Now find for the function when .

When , is positive, so the graph of the function is convex downwards. Also, when , is negative, so the graph of the function is convex upwards. This means that the graph of the function must have a point of inflection at . To find the -coordinate of the point, substitute 6 for in the equation and solve for .

Thus, the point of inflection of the function is .

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