Questions: Consider the function: f(x) = (9x² + 25) / x Step 1. Find all critical points of the function. Separate multiple answers with cor

Find the critical points of the function \( f(x) = \frac{9x^2 + 25}{x} \).

Derivative of the function

The function is given by \( f(x) = \frac{9x^2 + 25}{x} = 9x + \frac{25}{x} = 9x + 25x^{-1} \). The derivative is:
\[
f'(x) = 9 - 25x^{-2} = 9 - \frac{25}{x^2}
\]

Solve for \( f'(x) = 0 \)

Set the derivative equal to zero:
\[
9 - \frac{25}{x^2} = 0
\]
\[
9 = \frac{25}{x^2}
\]
\[
9x^2 = 25
\]
\[
x^2 = \frac{25}{9}
\]
\[
x = \pm \sqrt{\frac{25}{9}} = \pm \frac{5}{3}
\]
Thus, the critical points are \( x = \frac{5}{3} \) and \( x = -\frac{5}{3} \).

Check where \( f'(x) \) is undefined

The derivative \( f'(x) \) is undefined when the denominator is zero:
\[
x^2 = 0
\]
\[
x = 0
\]
However, \( x = 0 \) is not in the domain of the original function \( f(x) \), so it is not a critical point.

The critical points are \(\boxed{x = -\frac{5}{3}, \frac{5}{3}}\).

The critical points of the function are \(\boxed{x = -\frac{5}{3}, \frac{5}{3}}\).