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To solve part (c) of the problem, we need to find the value of \( p \) for which the vertex of the function \( f_p(x) = \frac{1}{2}x^2 + px + 4 \) lies on the line \( l: y = -3x + 8 \).

### Step 1: Determine the vertex of \( f_p(x) \)
The vertex form of a quadratic function \( ax^2 + bx + c \) has its vertex at 
\[ x = -\frac{b}{2a}. \]
For the function \( f_p(x) = \frac{1}{2}x^2 + px + 4 \), we have \( a = \frac{1}{2} \) and \( b = p \). Thus, the \( x \)-coordinate of the vertex is:
\[ x = -\frac{p}{2 \cdot \frac{1}{2}} = -p. \]

The \( y \)-coordinate of the vertex can be found by substituting \( x = -p \) into the function:
\[ y = f_p(-p) = \frac{1}{2}(-p)^2 + p(-p) + 4 = \frac{1}{2}p^2 - p^2 + 4 = \frac{1}{2}p^2 - p^2 + 4 = -\frac{1}{2}p^2 + 4. \]

Thus, the vertex of \( f_p(x) \) is:
\[ \left( -p, -\frac{1}{2}p^2 + 4 \right). \]

### Step 2: Set the vertex on line \( l: y = -3x + 8 \)
We need the vertex \( \left( -p, -\frac{1}{2}p^2 + 4 \right) \) to lie on the line \( y = -3x + 8 \). This means substituting \( x = -p \) and \( y = -\frac{1}{2}p^2 + 4 \) into the line equation:

\[
-\frac{1}{2}p^2 + 4 = -3(-p) + 8
\]

Simplify the right side:

\[
-\frac{1}{2}p^2 + 4 = 3p + 8
\]

### Step 3: Solve for \( p \)
Rearrange the equation to:

\[
-\frac{1}{2}p^2 - 3p + 4 - 8 = 0
\]
\[
-\frac{1}{2}p^2 - 3p - 4 = 0
\]
Multiply by -2 to clear the fraction:
\[
p^2 + 6p + 8 = 0
\]

This is a standard quadratic equation. We can solve it using the quadratic formula:

\[
p = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

where \( a = 1 \), \( b = 6 \), and \( c = 8 \). Thus:

\[
p = \frac{-6 \pm \sqrt{6^2 - 4 \cdot 1 \cdot 8}}{2 \cdot 1}
\]
\[
p = \frac{-6 \pm \sqrt{36 - 32}}{2}
\]
\[
p = \frac{-6 \pm \sqrt{4}}{2}
\]
\[
p = \frac{-6 \pm 2}{2}
\]
\[
p = \frac{-6 + 2}{2} = \frac{-4}{2} = -2
\]
\[
p = \frac{-6 - 2}{2} = \frac{-8}{2} = -4
\]

### Solution
The values of \( p \) for which the vertex of \( f_p(x) \) lies on the line \( l: y = -3x + 8 \) are \( p = -2 \) and \( p = -4 \).

Would you like to explore further details or solve any related problems?

### Related Questions
1. For which value of \( p \) does \( f_p(x) \) have its vertex on the line \( y = kx + m \)?
2. How do you determine if a quadratic function has a positive minimum?
3. Can you derive the general formula for the vertex of a quadratic function?
4. How does changing the parameter \( p \) affect the graph of \( f_p(x) \)?
5. What is the relationship between the coefficients of a quadratic function and the line where its vertex lies?

**Tip:** When solving for specific conditions in a quadratic function, identifying the vertex form or completing the square can be very helpful.

Gegeven zijn de functies f_p(x) = 1/2x^2 + px + 4. Bereken algebraïsch voor welke p: (a) de functie f_p een positief minimum heeft; (b) de top van de grafiek van f_p op de lijn k: y = -5 ligt; (c) de top van de grafiek van f_p op de lijn l: y = -3x + 8 ligt; (d) Stel de formule op van deze kromme.

Solve for p in the quadratic function f_p(x) = 1/2x^2 + px + 4, determining when the function has a positive minimum and when its vertex lies on specific lines. Learn key algebraic methods to solve for vertex and quadratic properties.

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