# Video: Finding the Point of Inflection of a Function Involving Using the Product Rule with Logarithmic Functions If Any

Find (if any) the inflection points of π(π₯) = 3π₯Β² ln 2π₯.

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### Video Transcript

Find, if any, the inflection points of π of π₯ equals three π₯ squared times the natural log of two π₯.

To find the points of inflection, weβll evaluate the second derivative of our function and set it equal to zero. Notice that our function is itself the product of two functions. So weβll need to use the product rule to differentiate it. This says that for two differentiable functions, π’ and π£, the derivative of their product is π’ times dπ£ by dπ₯ plus π£ times dπ’ by dπ₯. We let π’ be equal to three π₯ squared and π£ be equal to the natural log of two π₯. Then dπ’ by dπ₯ is six π₯ and dπ£ by dπ₯ is one over π₯. So π prime the first derivative of our function is three π₯ squared times one over π₯ plus six π₯ times the natural log of two π₯ or three π₯ plus six π₯ times the natural log of two π₯. Weβre going to differentiate this again.

We use the product rule to find that the derivative of six π₯ times the natural log of two π₯ is six plus six times the natural log of two π₯. And the second derivative is nine plus six times the natural log of two π₯. Letβs set this equal to zero. To solve for π₯, we subtract nine and then divide through by six. We raised both sides as the power of π. And then we divide through by two. So potentially, thereβs an inflection point at π₯ equals one-half π to the negative three over two. But we ought to check whether this is actually an inflection point by checking the values of π double prime or the second derivative to either side of this.

A half π to the negative three over two is approximately 0.112. So letβs try π₯ equals 0.1 and π₯ equals 0.12. The second derivative π double prime of 0.1 is less than zero. And π double prime of 0.12 is greater than zero. The curve goes from being concave down to being concave up. And we can say we do indeed have a point of inflection at π₯ equals one-half π to the negative three over two. Substituting this value of π₯ into original function, we get negative nine over eight π cubed. π has an inflection point at π to the negative three over two over two, negative nine over eight π cubed.