We are tasked with finding any local maximum and/or minimum values of the function:

$y = x^3 - 9x^2 + 15x + 5$

Steps to find local extrema:

1. First Derivative (Critical Points): To find the critical points, we first take the derivative of the function and set it equal to zero.

$y' = \frac{d}{dx} (x^3 - 9x^2 + 15x + 5) = 3x^2 - 18x + 15$

Set the derivative equal to zero to find the critical points:

$3x^2 - 18x + 15 = 0$

Simplify:

$x^2 - 6x + 5 = 0$

Factor the quadratic:

$(x - 5)(x - 1) = 0$

So, the critical points are $x = 5$ and $x = 1$.

2. Second Derivative (Concavity Test): To determine if these critical points are maxima or minima, we take the second derivative.

$y'' = \frac{d}{dx} (3x^2 - 18x + 15) = 6x - 18$

Now, evaluate the second derivative at the critical points:

• At $x = 1$:

$y''(1) = 6(1) - 18 = -12 \quad (\text{concave down, local maximum at } x = 1)$

• At $x = 5$:

$y''(5) = 6(5) - 18 = 12 \quad (\text{concave up, local minimum at } x = 5)$

Local Maximum and Minimum:

• Local maximum at $x = 1$. To find the value of $y$ at $x = 1$:

$y(1) = 1^3 - 9(1)^2 + 15(1) + 5 = 1 - 9 + 15 + 5 = 12$

Thus, the local maximum is at $(1, 12)$.

• Local minimum at $x = 5$. To find the value of $y$ at $x = 5$:

$y(5) = 5^3 - 9(5)^2 + 15(5) + 5 = 125 - 225 + 75 + 5 = -20$

Thus, the local minimum is at $(5, -20)$.

Sketch of the Function:

The graph of the function $y = x^3 - 9x^2 + 15x + 5$ has a local maximum at $(1, 12)$ and a local minimum at $(5, -20)$. The function tends to $+\infty$ as $x \to \infty$ and $-\infty$ as $x \to -\infty$, since it's a cubic function.

I will now create a rough sketch of this function for you.

Summary:

• Local maximum at $(1, 12)$
• Local minimum at $(5, -20)$

Let me know if you would like further clarifications or the image of the graph.

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Relative Questions:

1. What is the significance of the second derivative in determining concavity?
2. How would a function with a local maximum and minimum differ if it was a quadratic rather than cubic?
3. What is the behavior of a cubic function as $x \to \infty$ and $x \to -\infty$?
4. Can there be more than one local maximum or minimum for a cubic function? Why or why not?
5. How would we find inflection points for a cubic function?

Tip: Always check both the first and second derivatives to confirm the nature of critical points (maximum, minimum, or neither).