Video: Finding the Inflection Point on the Curve of a Quadratic Function If It Exists

Video Transcript

Determine the inflection points of the curve 𝑦 equals 𝑥 squared plus two 𝑥 minus five.

First of all, let’s remind ourselves what an inflection point is. It’s a point where the concavity changes. We go from concave up to concave down or concave down to concave up. To identify inflection points, we use the second derivative. So, let’s visualize why the second derivative helps us to find inflection points.

If this is a graph of a function 𝑓, not necessarily our function, we can see that this part of the graph is concave down and this part is concave up. So, the point in-between is an inflection point. Where the graph is concave down, the slope is decreasing. And where the graph is concave up, the slope is increasing. So, if we graph the derivative of this function 𝑓 prime, which is the slope function, it would look like this. Decreasing, while the graph of 𝑓 is concave down. And after the point of inflection, it’s increasing where the graph of 𝑓 is concave up.

What about the second derivative? Well, the slope of the graph of 𝑓 prime of 𝑥 is negative before the point of inflection and it’s positive after. So, what does this tell us? A point of inflection occurs where the second derivative changes sign from negative to positive or positive to negative. So, that’s why the second derivative helps us to find inflection points. Let’s go ahead and find our second derivative.

To do this, let’s remember the general power rule that the derivative of 𝑎𝑥 to the power of 𝑛 is 𝑛𝑎𝑥 to the power of 𝑛 minus one. So, for our function of 𝑥, 𝑥 squared plus two 𝑥 minus five, 𝑓 prime of 𝑥 equals two 𝑥 plus two. That’s because 𝑥 on its own is the same as 𝑥 to the power of one. So, two 𝑥 differentiates to two. And constants like negative five differentiate to zero. And now, we differentiate again to get our second derivative. This gives us 𝑓 double prime of 𝑥 equals two.

Remember, we said that, to find inflection points, we’re looking for points where the second derivative function changes sign. But we got a positive constant, so what does this mean? Well, 𝑓 of 𝑥 is concave up where the second derivative is positive. As the second derivative is a constant, it can’t change sign. So, the graph of 𝑦 must always be concave up. So, as the concavity doesn’t change, we conclude that the curve has no inflection points. In fact, if we draw a sketch of this function, we can see that the curve is always concave up. There are no points of inflection on this curve.