Linear Algebra

Find a Quadratic Function Satisfying Conditions on Derivatives

Problem 650

Find a quadratic function $f(x) = ax^2 + bx + c$ such that $f(1) = 3$, $f'(1) = 3$, and $f^{\prime\prime}(1) = 2$.

Here, $f'(x)$ and $f^{\prime\prime}(x)$ denote the first and second derivatives, respectively.

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Solution.

Each condition required on $f$ can be turned into an equation involving the constants $a, b, c$.

In particular, $f(1) = 3$ tells us that $a + b + c = 3$.

Because $f'(x) = 2ax + b$, the condition $f'(1) = 3$ gives us $2a + b = 3$.

And finally $f^{\prime\prime}(x) = 2a$, and so $f^{\prime\prime}(1) = 2a = 2$. Thus we have the system of equations
\begin{align*}
a + b + c &= 3 \\
2a + b &= 3\\
2a &= 2
\end{align*}

To solve this system, we could create the augmented matrix and then reduce it.

For this system, though, it is simpler to solve directly. The third equation tells us that $a=1$.

Plugging this value into the second equation, we find $b=1$.

Plugging both of these values into the first equation, we see $c=1$ as well.

Thus the function we want is $f(x) = x^2 + x + 1$.

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