Let's Learn by Doing
Let $f(x) = \frac{x + 10}{x^2 - 64}$. Find the critical numbers of $f$.
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Step 1
The critical numbers occur where $f'(x)$ is zero or undefined. So, we begin by calculating the derivative of
f.
f'(x) =
Step 2
Since critical numbers occur where $f'(x)$ is undefined, we need to determine the $x$ values where the
denominator, $(x^2 - 64)^2$, of $f'$ is undefined.
x =
Step 3
Critical numbers also exist where $f'(x) = 0$. And a rational function is zero when the numerator is zero
and the denominator is not zero. So, we need to find the values of $x$ where the numerator,
$-(x^2 + 20x + 64)$, of $f'$ is zero.
x =
Step 4
Therefore, the critical numbers of $f(x) = \frac{x + 10}{x^2 - 64}$ are:
