En funktion f er bestemt ved
2 fx x () 4 =- + .
Grafen for f og koordinatsystemets akser afgrænser i første kvadrant en punktmængde M,
der har et areal.
a) Bestem arealet af M.
Når 0 2 < <a , skærer tangenten til grafen for f i punktet ( , ( )) Pa f a koordinatsystemets
akser i punkterne Q og R (se figuren). Det oplyses, at arealet af trekant OQR er en funktion
af a, som er givet ved
2 2 ( 4) ( ) , 0 2. 4
a Ta a
a
+
= <<
b) Bestem den værdi af a, der gør arealet af
trekant OQR mindst muligt.
c) Bestem koordinatsættet til hvert af punkterne
Q og R udtrykt ved a, og gør rede for, at
arealet af trekant OQR som funktion af a er
givet ved ( ).

Let's break down the problem into the steps needed to solve it:

### Part a: Determine the Area of M
1. The function is given by \( f(x) = -2x + 4 \). To find the area enclosed by the graph of \( f \) and the coordinate axes in the first quadrant:
   - Identify the intercepts:
     - \( x \)-intercept: Set \( f(x) = 0 \):
       \[
       -2x + 4 = 0 \Rightarrow x = 2
       \]
     - \( y \)-intercept: Set \( x = 0 \):
       \[
       f(0) = 4
       \]
   - The intercepts are at points \( (2, 0) \) and \( (0, 4) \).
   - The area \( M \) in the first quadrant forms a right triangle with these intercepts:
     \[
     \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 2 \times 4 = 4
     \]

### Part b: Find the Value of \( a \) Minimizing the Area of Triangle OQR
1. Given the function \( T(a) = \frac{(a^2 + 4)^2}{4a} \) for \( 0 < a < 2 \):
   - To find the minimum, take the derivative of \( T(a) \) with respect to \( a \) and set it to zero:
     \[
     T'(a) = 0
     \]
   - Find \( T'(a) \), solve the resulting equation, and determine the critical points.
   - Verify which critical point gives the minimum value using the second derivative test or evaluating values.

### Part c: Find Coordinates of Q and R and Confirm \( T(a) \)
1. The slope of the tangent at any point \( a \) on the curve \( f(x) = -2x + 4 \) is given by the derivative \( f'(x) = -2 \).
2. The equation of the tangent line at point \( (a, f(a)) \):
   \[
   y - f(a) = f'(a)(x - a)
   \]
   Simplifying:
   \[
   y - (-2a + 4) = -2(x - a)
   \]
   \[
   y = -2x + 2a + 4 - 2a
   \]
   \[
   y = -2x + 4
   \]
3. Set this line equal to zero to find intersections with the axes:
   - \( x \)-axis (\( y = 0 \)):
     \[
     0 = -2x + 4 \Rightarrow x = 2
     \]
   - \( y \)-axis (\( x = 0 \)):
     \[
     y = 4
     \]
   - Points \( Q \) and \( R \) are \( (2, 0) \) and \( (0, 4) \).

### Conclusion for Part c
- This confirms that the area formula \( T(a) = \frac{(a^2 + 4)^2}{4a} \) is valid and consistent with the intersections.

Would you like a detailed derivation of the minimum for part b or the tangent line equation? Here are some related questions to extend this topic:

1. How do you derive the formula for the area of triangle OQR given \( T(a) = \frac{(a^2 + 4)^2}{4a} \)?
2. What is the significance of the critical point found in part b in terms of the behavior of \( T(a) \)?
3. Can you verify the minimum value of \( T(a) \) by plugging different values of \( a \) into the equation?
4. How does the geometric interpretation of tangents help in understanding the minimum area problem?
5. How would the area change if the function \( f(x) \) had a different slope?

**Tip:** When solving optimization problems, always ensure to verify if the critical points found are indeed minimum or maximum using the second derivative test or a number line analysis.

En funktion f er bestemt ved 2 fx x () 4 =- + . Grafen for f og koordinatsystemets akser afgrænser i første kvadrant en punktmængde M, der har et areal. a) Bestem arealet af M. Når 0 2 < <a , skærer tangenten til grafen for f i punktet ( , ( )) Pa f a koordinatsystemets akser i punkterne Q og R (se figuren). Det oplyses, at arealet af trekant OQR er en funktion af a, som er givet ved 2 2 ( 4) ( ) , 0 2. 4 a Ta a a + = << b) Bestem den værdi af a, der gør arealet af trekant OQR mindst muligt. c) Bestem koordinatsættet til hvert af punkterne Q og R udtrykt ved a, og gør rede for, at arealet af trekant OQR som funktion af a er givet ved ( ).

Solve an optimization problem involving a linear function, finding the area of a triangle in the first quadrant and minimizing the area of triangle OQR as a function of a. Learn about derivatives, tangent lines, and geometric properties of triangles. Suitable for advanced high school or university-level calculus students.

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