Consider a function f(r) = axr2 t br t c. Show that the Taylor's approximation of first order of f(r) at any point xo is always greater than or always less than f(r) for any value of r. How can you decide just by examining the quadratic whether the Taylor's approximation of first order will be greater than or less than f(r) for any x?

Given the function \( f(r) = ar^2 + br + c \), we want to show that the first-order Taylor approximation of \( f(r) \) at any point \( r_0 \) is always greater than or always less than \( f(r) \) for any value of \( r \). Moreover, we want to determine how to decide this by examining the quadratic form of the function.

### Step 1: Calculate the First-Order Taylor Approximation
The first-order Taylor approximation of \( f(r) \) at \( r = r_0 \) is given by:
\[
T_1(r) = f(r_0) + f'(r_0)(r - r_0)
\]
where:
- \( f'(r) = \frac{d}{dr} (ar^2 + br + c) \) is the first derivative of \( f(r) \).

### Step 2: Compute the Derivative \( f'(r) \)
The first derivative of \( f(r) \) with respect to \( r \) is:
\[
f'(r) = 2ar + b
\]

So, the first-order Taylor approximation at \( r = r_0 \) is:
\[
T_1(r) = f(r_0) + (2ar_0 + b)(r - r_0)
\]

### Step 3: Compare \( f(r) \) and \( T_1(r) \)
The original function \( f(r) \) can be rewritten as:
\[
f(r) = ar^2 + br + c
\]
Now, let’s express the difference between \( f(r) \) and \( T_1(r) \):
\[
f(r) - T_1(r) = (ar^2 + br + c) - \left[ f(r_0) + (2ar_0 + b)(r - r_0) \right]
\]

### Step 4: Simplify the Difference
First, note that \( f(r_0) = ar_0^2 + br_0 + c \). Substituting this into the expression for \( T_1(r) \), we get:
\[
f(r) - T_1(r) = ar^2 + br + c - \left[ ar_0^2 + br_0 + c + (2ar_0 + b)(r - r_0) \right]
\]

Simplify the expression:
\[
f(r) - T_1(r) = ar^2 + br + c - ar_0^2 - br_0 - c - (2ar_0 + b)(r - r_0)
\]

Further simplification yields:
\[
f(r) - T_1(r) = a(r^2 - r_0^2) + b(r - r_0) - (2ar_0 + b)(r - r_0)
\]

Notice that \( r^2 - r_0^2 = (r - r_0)(r + r_0) \), so we can substitute this:
\[
f(r) - T_1(r) = a(r - r_0)(r + r_0) + b(r - r_0) - (2ar_0 + b)(r - r_0)
\]

Factoring \( (r - r_0) \) out:
\[
f(r) - T_1(r) = (r - r_0)\left[a(r + r_0) + b - 2ar_0 - b \right]
\]
\[
f(r) - T_1(r) = (r - r_0)\left[a(r + r_0 - 2r_0) \right]
\]
\[
f(r) - T_1(r) = a(r - r_0)(r - r_0)
\]
\[
f(r) - T_1(r) = a(r - r_0)^2
\]

### Step 5: Analyze the Sign of \( f(r) - T_1(r) \)
The expression \( f(r) - T_1(r) = a(r - r_0)^2 \) is always non-negative since \( (r - r_0)^2 \) is a square term and therefore always non-negative.

- If \( a > 0 \), then \( f(r) - T_1(r) \geq 0 \), meaning \( f(r) \geq T_1(r) \). In this case, the Taylor approximation is always less than or equal to the actual function \( f(r) \).
- If \( a < 0 \), then \( f(r) - T_1(r) \leq 0 \), meaning \( f(r) \leq T_1(r) \). In this case, the Taylor approximation is always greater than or equal to the actual function \( f(r) \).

### Conclusion
The first-order Taylor approximation will be less than the function \( f(r) \) for all \( r \) if \( a > 0 \), and it will be greater than the function \( f(r) \) for all \( r \) if \( a < 0 \). This conclusion is determined solely by the sign of the quadratic coefficient \( a \).

### Follow-Up Questions
1. How does the sign of \( a \) influence the concavity of the function \( f(r) \)?
2. What happens to the approximation if \( a = 0 \)?
3. How would the second-order Taylor approximation compare to the function \( f(r) \)?
4. Can you derive a similar conclusion for a cubic polynomial?
5. How does the location of \( r_0 \) affect the accuracy of the Taylor approximation?

### Tip
Remember, the sign of the leading coefficient in a quadratic function determines whether the parabola opens upwards or downwards, which directly influences whether the Taylor approximation underestimates or overestimates the function.

Explore how to determine whether the first-order Taylor approximation of a quadratic function will be greater than or less than the function itself. This detailed mathematical analysis covers the Taylor series, first-order approximation formulas, and the quadratic function properties. Suitable for college-level students and enthusiasts of mathematical analysis.

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