If 𝑓 of 𝑥 is equal to 𝑥 lots of 𝑎 minus ln of 𝑥 such that 𝑎 is a constant and the curve of the function has a critical point at 𝑥 equals 𝑒, find 𝑎.
First, let’s remember what a critical point is. A critical point is a point, such as 𝑥 is equal to 𝑐, such that the differential of 𝑓 at 𝑐, which we can also call 𝑓 dash of 𝑐, is equal to zero. So let’s start answering this question by differentiating 𝑓 of 𝑥 with respect to 𝑥. And this is also called 𝑓 dash of 𝑥. And it’s equal to d by d𝑥 of 𝑥 times by 𝑎 minus ln of 𝑥. In order to differentiate this, we’ll need to use the product rule. The product rule tells us that if we have some function 𝑦, which is equal to a product such as 𝑢 multiplied by 𝑣, then when we differentiate 𝑦, we’ll get d𝑦 by d𝑥 is equal to 𝑢 d𝑣 by d𝑥 plus 𝑣 d𝑢 by d𝑥.
Now, in our case, we can let 𝑥 be 𝑢 and let 𝑎 minus ln of 𝑥 be 𝑣, then we obtain that this differential is equal to 𝑥 times by d by d𝑥 of 𝑎 minus ln of 𝑥 plus 𝑎 minus ln of 𝑥 times d by d𝑥 of 𝑥. And, immediately, we can evaluate d by d𝑥 of 𝑥, since this is just equal to one. In order to differentiate 𝑎 minus ln of 𝑥 with respect to 𝑥, we’ll need to use a rule. And this rule tells us that d by d𝑥 of ln of 𝑥 is equal to one over 𝑥. Now, in order to answer this question, you do not need to know how to derive this differential. But for those of you who are interested, I shall write down the derivation of this differential. And you can pause the video and read through how it’s done.
So this is how you can prove that d by d𝑥 of ln of 𝑥 is equal to one over 𝑥. However, it’s not necessary to know this in order to answer the question. All you need to remember is that the differential of the natural logarithm of 𝑥 is equal to one over 𝑥. So now we can use this in order to resolve the differential of 𝑎 minus ln of 𝑥. Now, we’re told in the question that 𝑎 is a constant. And any constant will differentiate to zero. So the 𝑎 will go to zero, and then we have minus ln of 𝑥. And since ln of 𝑥 differentiates to one over 𝑥, negative ln of 𝑥 will differentiate to negative one over 𝑥. And then, we need to add on the 𝑎 minus ln of 𝑥 times by one.
Now, we can simplify this to give us that 𝑓 dash of 𝑥 is equal to negative one plus 𝑎 minus ln of 𝑥. Now that we found 𝑓 dash of 𝑥, we need to use the fact that this function has a critical point at 𝑥 equals 𝑒. If we remember back to the definition, if a critical point of a function is at 𝑥 equals 𝑐, then this means that 𝑓 dash of 𝑐 is equal to zero. Since our critical point is at 𝑒, we need to substitute 𝑥 equals 𝑒 into this equation and set it equal to zero.
This gives us that 𝑓 dash of 𝑒, which is equal to negative one plus 𝑎 minus ln of 𝑒, is equal to zero. Now, since ln is the natural logarithm or log to base 𝑒, ln of 𝑒 is simply one. And, therefore, our equation becomes 𝑎 minus two is equal to zero. This can be arranged for 𝑎 to give us that 𝑎 is equal to two. And this is our solution.