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.