Key Concepts and Formulas

• Derivative of a Quadratic Function: If $f(x) = ax^2 + bx + c$, then $f'(x) = 2ax + b$.
• Arithmetico-Geometric Progression (AGP): A sequence where each term is the product of corresponding terms of an arithmetic progression (AP) and a geometric progression (GP). If the AP terms are $a, a+d, a+2d, \dots$ and GP terms are $y, yr, yr^2, \dots$, then the AGP terms are $ay, (a+d)yr, (a+2d)yr^2, \dots$.
• Solving Systems of Equations: Using substitution or elimination to find the values of unknown variables.

Step-by-Step Solution

Step 1: Understand the structure of the coefficients $a_0, a_1, a_2$ as an AGP. We are given that $a_0, a_1, a_2$ are in an Arithmetico-Geometric Progression. The corresponding AP has a common difference of 1, and the corresponding GP has a common ratio of 2. Let the first term of the AP be $a$ and the first term of the GP be $y$. The terms of the AP are: $a, a+1, a+2$. The terms of the GP are: $y, y \cdot 2, y \cdot 2^2$, which simplifies to $y, 2y, 4y$. Therefore, the terms of the AGP are: $a_0 = a \cdot y$ $a_1 = (a+1) \cdot 2y$ $a_2 = (a+2) \cdot 4y$

Step 2: Use the given derivative information to find relationships between $a_0$ and $a_1$. The function is given by $f(x) = a_0x^2 + a_1x + a_2$. The derivative of $f(x)$ is $f'(x) = 2a_0x + a_1$. We are given $f'(0) = 1$. Substituting $x=0$ into $f'(x)$: $f'(0) = 2a_0(0) + a_1 = a_1$ So, $a_1 = 1$.

We are also given $f'(1) = 0$. Substituting $x=1$ into $f'(x)$: $f'(1) = 2a_0(1) + a_1 = 2a_0 + a_1$ So, $2a_0 + a_1 = 0$.

Step 3: Solve for $a_0$ using the values of $a_1$. From Step 2, we have $a_1 = 1$ and $2a_0 + a_1 = 0$. Substitute $a_1 = 1$ into the second equation: $2a_0 + 1 = 0$ $2a_0 = -1$ $a_0 = -\frac{1}{2}$

Step 4: Determine the values of $a$ and $y$ from the AGP definitions and the calculated coefficients. From Step 1, we have $a_0 = ay$ and $a_1 = (a+1)2y$. From Step 3, we know $a_0 = -\frac{1}{2}$ and $a_1 = 1$. So, we have the following system of equations:

1. $ay = -\frac{1}{2}$
2. $(a+1)2y = 1$

Let's expand the second equation: $2ay + 2y = 1$ Now, substitute the value of $ay$ from equation (1) into this expanded equation: $2\left(-\frac{1}{2}\right) + 2y = 1$ $-1 + 2y = 1$ $2y = 2$ $y = 1$

Now, substitute the value of $y=1$ back into equation (1): $a(1) = -\frac{1}{2}$ $a = -\frac{1}{2}$

Step 5: Calculate the value of $a_2$ using its AGP definition. From Step 1, $a_2 = (a+2)4y$. Substitute the values $a = -\frac{1}{2}$ and $y = 1$ that we found in Step 4: $a_2 = \left(-\frac{1}{2} + 2\right) \cdot 4 \cdot 1$ $a_2 = \left(-\frac{1}{2} + \frac{4}{2}\right) \cdot 4$ $a_2 = \left(\frac{3}{2}\right) \cdot 4$ $a_2 = 6$

Step 6: Write the complete function $f(x)$ and evaluate $f(4)$. We have found the coefficients: $a_0 = -\frac{1}{2}$ $a_1 = 1$ $a_2 = 6$

So, the function $f(x)$ is: $f(x) = -\frac{1}{2}x^2 + 1x + 6$

Now, we need to find $f(4)$: $f(4) = -\frac{1}{2}(4)^2 + 1(4) + 6$ $f(4) = -\frac{1}{2}(16) + 4 + 6$ $f(4) = -8 + 4 + 6$ $f(4) = 2$

Common Mistakes & Tips

• Incorrectly setting up the AGP terms: Ensure the correct correspondence between AP and GP terms and their products.
• Algebraic errors in solving the system of equations: Double-check substitutions and simplifications, especially with fractions.
• Confusing coefficients with AP/GP terms: Clearly distinguish between $a_0, a_1, a_2$ and the underlying 'a' and 'y' of the AGP.

Summary

The problem requires understanding the definition of an Arithmetico-Geometric Progression and its relation to arithmetic and geometric progressions. We first utilized the derivative of the quadratic function $f(x)$ and the given conditions $f'(0)=1$ and $f'(1)=0$ to determine the values of the coefficients $a_0$ and $a_1$. Subsequently, we used the AGP property of the coefficients ($a_0, a_1, a_2$) and the given common difference of the AP (1) and common ratio of the GP (2) to set up equations involving the initial terms of the AP ($a$) and GP ($y$). Solving these equations allowed us to find $a$ and $y$, which in turn enabled us to calculate $a_2$. Finally, with all coefficients determined, we constructed the function $f(x)$ and evaluated it at $x=4$ to obtain the answer.

The final answer is \boxed{2}.