To find the local minimum and maximum of the function $f(x) = 6x + 9x^{-1}$, we need to determine the critical points by following these steps:

Step 1: Differentiate $f(x)$

The function is: $f(x) = 6x + \frac{9}{x}$

Taking the derivative $f'(x)$: $f'(x) = 6 - \frac{9}{x^2}$

Step 2: Set $f'(x) = 0$ to find critical points

$6 - \frac{9}{x^2} = 0$ Solving for $x$: $\frac{9}{x^2} = 6$ $x^2 = \frac{9}{6} = \frac{3}{2}$ $x = \pm \sqrt{\frac{3}{2}}$

Thus, the critical points are $x = \sqrt{\frac{3}{2}}$ and $x = -\sqrt{\frac{3}{2}}$.

Step 3: Determine if the critical points are maxima or minima

To classify these points, we can examine the second derivative $f''(x)$ or use the first derivative test.

The second derivative of $f(x)$ is: $f''(x) = \frac{18}{x^3}$

Evaluating $f''(x)$ at $x = \sqrt{\frac{3}{2}}$ and $x = -\sqrt{\frac{3}{2}}$:

1. For $x = \sqrt{\frac{3}{2}}$: $f''\left(\sqrt{\frac{3}{2}}\right) > 0$ This indicates a local minimum.

2. For $x = -\sqrt{\frac{3}{2}}$: $f''\left(-\sqrt{\frac{3}{2}}\right) < 0$ This indicates a local maximum.

Step 4: Calculate $f(x)$ at the critical points

1. Local Maximum at $x = -\sqrt{\frac{3}{2}}$: $f\left(-\sqrt{\frac{3}{2}}\right) = 6\left(-\sqrt{\frac{3}{2}}\right) + \frac{9}{-\sqrt{\frac{3}{2}}} = -12\sqrt{\frac{3}{2}}$

2. Local Minimum at $x = \sqrt{\frac{3}{2}}$: $f\left(\sqrt{\frac{3}{2}}\right) = 6\sqrt{\frac{3}{2}} + \frac{9}{\sqrt{\frac{3}{2}}} = 12\sqrt{\frac{3}{2}}$

Conclusion

• Local Maximum: $x = -\sqrt{\frac{3}{2}}$ with value $-12\sqrt{\frac{3}{2}}$.
• Local Minimum: $x = \sqrt{\frac{3}{2}}$ with value $12\sqrt{\frac{3}{2}}$.

Would you like further details or have any questions?

Here are five related questions for practice:

1. How do you determine the concavity of a function at a critical point?
2. What does the second derivative test indicate about a function's behavior?
3. How would the values of local maxima and minima change if the function had a different coefficient?
4. Can there be more than one local maximum or minimum in a polynomial of degree 3?
5. How can we interpret the behavior of $f(x)$ as $x \to 0$ or $x \to \infty$?

Tip: Checking the sign of $f''(x)$ at critical points quickly identifies local minima and maxima by indicating concavity.