Question Video: Finding the Unknowns in a Rational Function given Its Value and the Value of Its First Derivative at a Point Mathematics

Video Transcript

Suppose that 𝑓 of 𝑥 is equal to 𝑥 squared plus 𝑎𝑥 plus 𝑏 all over 𝑥 squared minus seven 𝑥 plus four. Given that 𝑓 of zero is equal to one and 𝑓 prime of zero is equal to four, find 𝑎 and 𝑏.

Our first step in this question can be to substitute 𝑥 equals zero into 𝑓 of 𝑥. Since we’re given that 𝑓 of zero is equal to one. We obtain that 𝑓 of zero is equal to zero squared plus 𝑎 times zero plus 𝑏 all over zero squared minus seven times zero plus four. Now, all of these terms will go to zero apart from 𝑏 and four. We’re left with 𝑓 of zero is equal to 𝑏 over four. Next, we use the fact that the question has told us that 𝑓 of zero is equal to one. And so, we can set this equal to one. From this, we find that 𝑏 is equal to four. Next, we can use the fact that 𝑓 prime of zero is equal to four. However, first of all, we must find 𝑓 prime of 𝑥. In order to do this, we need to differentiate 𝑓. Since 𝑓 is a rational function, we can use the quotient rule in order to find its derivative.

The quotient rule tells us that 𝑢 over 𝑣 prime is equal to 𝑣 times 𝑢 prime minus 𝑢 times 𝑣 prime all over 𝑣 squared. Setting our function 𝑓 of 𝑥 equal to 𝑢 over 𝑣, we obtain that 𝑢 is equal to 𝑥 squared plus 𝑎𝑥 plus 𝑏. And 𝑣 is equal to 𝑥 squared minus seven 𝑥 plus four. We can find 𝑢 prime and 𝑣 prime by differentiating these two functions. Giving us that 𝑢 prime is equal to two 𝑥 plus 𝑎 and 𝑣 prime is equal to two 𝑥 minus seven. Now, we can substitute these into the quotient rule. We obtain that 𝑓 prime of 𝑥 is equal to 𝑥 squared minus seven 𝑥 plus four multiplied by two 𝑥 plus 𝑎 minus 𝑥 squared plus 𝑎𝑥 plus 𝑏 multiplied by two 𝑥 minus seven all over 𝑥 squared minus seven 𝑥 plus four all squared.

Now, we could simplify 𝑓 prime of 𝑥 at this point. However, we’re going to be substituting in 𝑥 is equal to zero. And so, a lot of these terms would just disappear. Let’s simply substitute 𝑥 equals zero in here. We obtain this. However, a lot of the terms will vanish to zero, which leaves us with four 𝑎 plus seven 𝑏 all over 16. Now, we have found that 𝑏 is equal to four earlier. And so, we can substitute this in, giving us four 𝑎 plus 28 all over 16.

Since the question has told us that 𝑓 prime of zero is equal to four, we can set this equal to four. Then, we simply rearrange this in order to solve for 𝑎. Now, we obtain our solution that 𝑎 is equal to nine. We’ve now found the values of both 𝑎 and 𝑏, which completes the solution to this question.