Video Transcript
Let 𝑓 of 𝑥 be equal to 𝑥 cubed minus four 𝑥 minus 18. What value of 𝑥 solves the equation 𝑓 prime of 𝑥 is equal to 92?
The question gives us a cubic function 𝑓 of 𝑥. And it wants us to find the values of 𝑥 which solve the equation 𝑓 prime of 𝑥 is equal to 92. First, we need to remember that 𝑓 prime of 𝑥 means the derivative function of 𝑓 of 𝑥. So, to find 𝑓 prime of 𝑥, we need to differentiate our cubic equation 𝑥 cubed minus four 𝑥 minus 18.
We remember when we’re differentiating the sum or difference of terms, we can just differentiate each term separately. And we can differentiate all of these terms by using the power rule for differentiation. For any constants 𝑎 and 𝑛, the derivative of 𝑎𝑥 to the 𝑛th power with respect to 𝑥 is equal to 𝑛𝑎𝑥 to the power of 𝑛 minus one. We multiply by the exponent of 𝑥 and reduce this exponent by one.
So, we first differentiate 𝑥 cubed. We multiply by the exponent of three and reduce this exponent by one. And we can simplify this to three 𝑥 squared. Next, to differentiate our second term of negative four 𝑥, we’ll rewrite this as negative four 𝑥 to the first power. Now, to differentiate this, we multiply by the exponent of 𝑥, which is one, and then reduce this exponent by one. And we can again simplify this. Multiplying by one does not change our value. And 𝑥 to the power of one minus one is 𝑥 to the zeroth power. And 𝑥 to the zeroth power is equal to one. So, this expression simplifies to give us negative four.
We could differentiate our final term of negative 18 by rewriting it as negative 18 times 𝑥 to the zeroth power. However, we could also just remember the derivative of any constant is equal to zero. So, we found an expression for 𝑓 prime of 𝑥. It’s equal to three 𝑥 squared minus four. Remember, the question wants us to find the values of 𝑥 where 𝑓 prime of 𝑥 is equal to 92. So, we’ll set 𝑓 prime of 𝑥 to be 92. This gives us 92 is equal to three 𝑥 squared minus four.
We want to solve this equation for 𝑥. We’ll start by adding four to both sides of this equation. And this gives us 96 is equal to three 𝑥 squared. Now, we’ll divide through both sides of this equation by three. And this gives us that 32 is equal to 𝑥 squared. Finally, we can solve this equation for 𝑥 by taking the square root of both sides where we remember we’ll get a positive and a negative answer. This gives us that 𝑥 is equal to positive or negative root 32.
And we could leave our answer like this. However, we can simplify this by remembering that 32 is two to the fifth power, which we can write as two times two to the fourth power. Using this, we see the square root of 32 is the square root of two times two to the fourth power. Then, using our laws of exponents, we can rewrite this as the square of two times the square root of two to the fourth power. And two to the fourth power is equal to 16. So, the positive square root of 16 is equal to four. So, we can rewrite root 32 as four root two.
Therefore, we’ve shown if 𝑓 of 𝑥 is equal to 𝑥 cubed minus four 𝑥 minus 18 and 𝑓 prime of 𝑥 is equal to 92, then 𝑥 must be equal to four root two or 𝑥 must be equal to negative four root two.
