Question Video: Finding the Inflection Point of the Curve of a Polynomial Function | Nagwa Question Video: Finding the Inflection Point of the Curve of a Polynomial Function | Nagwa

Question Video: Finding the Inflection Point of the Curve of a Polynomial Function Mathematics • Third Year of Secondary School

Find the inflection point on the graph of 𝑓(𝑥) = 𝑥³ − 9𝑥² + 6𝑥.

03:43

Video Transcript

Find the inflection point on the graph of 𝑓 of 𝑥 equals 𝑥 cubed minus nine 𝑥 squared plus six 𝑥.

Remember, in order to find the inflection point on the graph of a function, we need to identify the point at which the convexity of that function changes. And whilst an inflection point might appear at a critical point, this is not necessarily the case. We can find the possible locations of one, though, by using the second derivative. Specifically, if the second derivative is equal to zero or does not exist, then inflection point might occur. To be sure, we can check whether the convexity either side of that point changes. In other words, does the graph change from being convex up to down or vice versa?

Now 𝑓 of 𝑥 is a polynomial. And this is great because it means it’s continuous and differentiable over its entire domain. In fact, when we differentiate 𝑓 of 𝑥, we get another polynomial, which is still continuous and differentiable. This means we can begin by finding the first derivative and then differentiate one more time to get the second derivative.

We differentiate term by term. The derivative of 𝑥 cubed with respect to 𝑥 is three 𝑥 squared. Remember, we multiply by the exponent and then reduce that the exponent by one. Similarly, differentiating negative nine 𝑥 squared with respect to 𝑥, and we get negative 18𝑥. Finally, the derivative of six 𝑥 is simply six. So the first derivative 𝑓 prime of 𝑥 is three 𝑥 squared minus 18𝑥 plus six.

We’ll repeat this process one more time to find the expression for the second derivative. This time, we get six 𝑥 minus 18. And to establish where an inflection point might occur, we’re going to set this equal to zero. In other words, six 𝑥 minus 18 is equal to zero. To solve for 𝑥, we begin by adding 18 to both sides, so 18 equals six 𝑥 or six 𝑥 equals 18. Then, we divide through by six. So we get 𝑥 equals 18 over six, or simply three. So we can deduce that an inflection point might occur at the point where 𝑥 equals three. This is the point where the second derivative is equal to zero.

To confirm that this is indeed an inflection point, we’ll look at the second derivative either side of 𝑥 equals three and check that the convexity does indeed change. Specifically, let’s check the points where 𝑥 equals two and 𝑥 equals four. Substituting 𝑥 equals two into our expression for the second derivative, and we get six times two minus 18, which is negative six. We know that if the second derivative is negative, the slope of the function is decreasing. This means over the interval where the second derivative is negative, the function is convex up. So our function is convex up at the point where 𝑥 equals two.

Repeating this for 𝑥 equals four, and we get that the second derivative is six times four minus 18, which is positive six. This time, 𝑓 of 𝑥 is convex down since the second derivative is positive at 𝑥 equals four. So we observe that the graph of the function changes from being convex up to convex down about the point 𝑥 equals three.

We can determine the full coordinates of this point by substituting 𝑥 equals three into the original function. 𝑓 of three is three cubed minus nine times three squared plus six times three, which is equal to negative 36. So the inflection point on the graph of our function is at three, negative 36.

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