Let’s solve the parts step-by-step:

(a) Find $f(-2)$

Given the function:
$f(x) = \frac{3}{4}x^4 - x^3 - 9x^2 + 20$ Substitute $x = -2$ into the function:

$f(-2) = \frac{3}{4}(-2)^4 - (-2)^3 - 9(-2)^2 + 20$

Simplify each term: $(-2)^4 = 16 \quad \text{so} \quad \frac{3}{4}(16) = 12$ $(-2)^3 = -8$ $(-2)^2 = 4 \quad \text{so} \quad 9(4) = 36$

Now plug everything into the expression: $f(-2) = 12 - (-8) - 36 + 20$ $f(-2) = 12 + 8 - 36 + 20 = 4$

Thus,
$f(-2) = 4$

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(b) Find $f'(x)$

We differentiate $f(x) = \frac{3}{4}x^4 - x^3 - 9x^2 + 20$.

Apply the power rule: $\frac{d}{dx}(x^n) = nx^{n-1}$.

$f'(x) = \frac{3}{4} \cdot 4x^3 - 3x^2 - 18x + 0$

Simplify: $f'(x) = 3x^3 - 3x^2 - 18x$

Thus,
$f'(x) = 3x^3 - 3x^2 - 18x$

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(c) Show that there is a second local minimum at $x = 3$

To check for local minima, we need to find the critical points by setting $f'(x) = 0$.

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

Factor out $3x$:

$3x(x^2 - x - 6) = 0$

Solve the quadratic inside the parentheses:

$x^2 - x - 6 = 0$

Factor:

$(x - 3)(x + 2) = 0$

So, the solutions are:

$x = 3 \quad \text{and} \quad x = -2$

These are the critical points. Since we are given that there is a local minimum at $x = -2$, we now check whether $x = 3$ also corresponds to a local minimum.

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Second Derivative Test

We use the second derivative to determine if the function has a local minimum at $x = 3$.

Start by finding $f''(x)$.
Differentiate $f'(x) = 3x^3 - 3x^2 - 18x$.

$f''(x) = 9x^2 - 6x - 18$

Now evaluate $f''(x)$ at $x = 3$:

$f''(3) = 9(3)^2 - 6(3) - 18$

Simplify:

$f''(3) = 9(9) - 18 - 18 = 81 - 36 = 45$

Since $f''(3) = 45 > 0$, the function is concave up at $x = 3$. Therefore, there is a local minimum at $x = 3$.

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Final Answer Summary

• (a) $f(-2) = 4$
• (b) $f'(x) = 3x^3 - 3x^2 - 18x$
• (c) There is a second local minimum at $x = 3$ since $f''(3) = 45 > 0$, indicating concave up.

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Do you have any questions or need further details?
Here are five related questions to explore:

1. What is the value of $f(3)$?
2. How would the function behave as $x \to \infty$ or $x \to -\infty$?
3. Can you graph the function to visualize the local minima and maxima?
4. How do you find points of inflection for the function?
5. What are the x-intercepts of the original function $f(x)$?

Tip: Always verify the sign of the second derivative at critical points to determine if it’s a local minimum or maximum.