To determine whether the first-order Taylor approximation of the function $f(x) = ax^2 + bx + c$ at any point $x_0$ is always greater than or always less than $f(x)$ for any value of $x$, we start by computing the first-order Taylor approximation and analyzing the quadratic function's properties.

1. First-Order Taylor Approximation

The first-order Taylor approximation of $f(x)$ at a point $x_0$ is given by:

$T_1(x) = f(x_0) + f'(x_0)(x - x_0)$

Where:

• $f(x_0) = ax_0^2 + bx_0 + c$
• $f'(x) = \frac{d}{dx}(ax^2 + bx + c) = 2ax + b$

Thus,

$T_1(x) = (ax_0^2 + bx_0 + c) + (2ax_0 + b)(x - x_0)$

2. Difference Between $f(x)$ and $T_1(x)$

We want to analyze the difference $D(x) = f(x) - T_1(x)$:

$D(x) = \left(ax^2 + bx + c\right) - \left[ax_0^2 + bx_0 + c + (2ax_0 + b)(x - x_0)\right]$

Simplifying $D(x)$:

$D(x) = ax^2 + bx + c - \left(ax_0^2 + bx_0 + c + (2ax_0 + b)(x - x_0)\right)$

Distribute and combine like terms:

$D(x) = ax^2 + bx + c - ax_0^2 - bx_0 - c - (2ax_0 + b)(x - x_0)$

$D(x) = ax^2 + bx - ax_0^2 - bx_0 - 2ax_0(x - x_0) - b(x - x_0)$

$D(x) = ax^2 - ax_0^2 - 2ax_0(x - x_0)$

$D(x) = a(x^2 - x_0^2 - 2x_0(x - x_0))$

$D(x) = a(x^2 - 2x_0x + x_0^2) = a(x - x_0)^2$

3. Analysis of $D(x) = a(x - x_0)^2$

The expression $D(x) = a(x - x_0)^2$ reveals whether the first-order Taylor approximation is greater than or less than $f(x)$ at any point $x_0$:

• $(x - x_0)^2$ is always non-negative.
• If $a > 0$, then $D(x) = a(x - x_0)^2 \geq 0$, meaning $T_1(x) \leq f(x)$ for all $x$. The Taylor approximation underestimates the function.
• If $a < 0$, then $D(x) = a(x - x_0)^2 \leq 0$, meaning $T_1(x) \geq f(x)$ for all $x$. The Taylor approximation overestimates the function.

Conclusion

The first-order Taylor approximation $T_1(x)$ is always greater than $f(x)$ when $a < 0$ and always less than $f(x)$ when $a > 0$. The sign of the coefficient $a$ of the quadratic term determines the relationship between the Taylor approximation and the actual function.

Summary

• If $a > 0$, $T_1(x)$ is less than $f(x)$ for all $x$.
• If $a < 0$, $T_1(x)$ is greater than $f(x)$ for all $x$.

Would you like further details or have any questions?

Here are 5 related questions you might find interesting:

1. How does the second-order Taylor approximation compare with the original function $f(x)$?
2. What happens to the Taylor approximation if $a = 0$?
3. How can you extend this analysis to functions with higher degrees, such as cubic functions?
4. Can you find the general form for the error term in the Taylor approximation?
5. How does the concavity of the function relate to the Taylor approximation?

Tip: The sign of the leading coefficient in a polynomial provides crucial information about the function's curvature and behavior, which is essential for understanding approximations.