To determine the local maximum and minimum values of the function

$f(x) = -x^3 + 9x^2 + 3,$

using the second derivative test, we need to follow these steps:

Step 1: Find the first derivative of $f(x)$

The first derivative $f'(x)$ gives us the slope of the function, and critical points occur where $f'(x) = 0$.

$f'(x) = \frac{d}{dx} \left( -x^3 + 9x^2 + 3 \right)$

Applying the power rule to each term:

$f'(x) = -3x^2 + 18x.$

Step 2: Find the critical points

To find the critical points, set $f'(x) = 0$ and solve for $x$:

$-3x^2 + 18x = 0$

Factor the equation:

$-3x(x - 6) = 0$

This gives two solutions:

$x = 0 \quad \text{or} \quad x = 6.$

So, the critical points are $x = 0$ and $x = 6$.

Step 3: Find the second derivative of $f(x)$

The second derivative $f''(x)$ helps us determine the concavity of the function and apply the second derivative test:

$f''(x) = \frac{d}{dx} \left( -3x^2 + 18x \right)$

Taking the derivative of each term:

$f''(x) = -6x + 18.$

Step 4: Apply the second derivative test

Now, we evaluate $f''(x)$ at each critical point.

• For $x = 0$:

$f''(0) = -6(0) + 18 = 18.$

Since $f''(0) > 0$, the function is concave up at $x = 0$, which means that $x = 0$ is a local minimum.

• For $x = 6$:

$f''(6) = -6(6) + 18 = -36 + 18 = -18.$

Since $f''(6) < 0$, the function is concave down at $x = 6$, which means that $x = 6$ is a local maximum.

Step 5: Find the function values at the critical points

• For $x = 0$:

$f(0) = -(0)^3 + 9(0)^2 + 3 = 3.$

So, the local minimum value is $f(0) = 3$.

• For $x = 6$:

$f(6) = -(6)^3 + 9(6)^2 + 3 = -216 + 324 + 3 = 111.$

So, the local maximum value is $f(6) = 111$.

Final Answer:

• The local minimum value is $f(0) = 3$ at $x = 0$.
• The local maximum value is $f(6) = 111$ at $x = 6$.

Would you like further details or explanations?

Here are some related questions:

1. How do you find the critical points of a function?
2. What does the second derivative test tell us about concavity?
3. How do you determine whether a function has an absolute maximum or minimum?
4. What are inflection points, and how do they relate to concavity?
5. Can the second derivative test fail, and if so, how?

Tip: Always check if the second derivative test applies, and if it gives an inconclusive result (i.e., $f''(x) = 0$), you may need to use the first derivative test instead.