Question

Find a second-degree polynomial \(P\) such that \(P\left( 2 \right) = 5\), \(P'\left( 2 \right) = 3\), and \(P''\left( 2 \right) = 2\).

Short answer

The required second-degree polynomial is \(P\left( x \right) = {x^2} - x + 3\).

Step-by-step solution

Differentiation of the Function

Differentiation is a process in which any function with one or more than one variable undergoes a calculation to determine the behavior of the curve of that function.

Evaluating Derivative.

Heregiven a second-degree polynomial \(P\) such that \(P\left( 2 \right) = 5\), \(P'\left( 2 \right) = 3\), and \(P''\left( 2 \right) = 2\).

Let there be a polynomial \(P\) such that \(P\left( x \right) = a{x^2} + bx + c\).

Differentiating with respect to\(x\)as:

\(\begin{align}\frac{{dP}}{{dx}} &= \frac{d}{{dx}}\left( {a{x^2} + bx + c} \right)\\P'\left( x \right) &= 2ax + b\\P''\left( x \right) &= 2a\end{align}\)

Computation in terms of given data.

Now,

\(\begin{align}P''\left( 2 \right) &= 2\\2a &= 2\\a &= 1\end{align}\)

Similarly,

\(\begin{align}P'\left( 2 \right) &= 3\\2a\left( 2 \right) + b &= 3\\4 + b &= 3\\b &= - 1\end{align}\)

And,

\(\begin{align}P\left( 2 \right) &= 5\\\left( 1 \right){\left( 2 \right)^2} + \left( { - 1} \right)\left( 2 \right) + c &= 5\\4 - 2 + c &= 5\\c &= 3\end{align}\)

So, the second-degree polynomial can be given as:

\(\begin{align}P\left( x \right) &= a{x^2} + bx + c\\ &= {x^2} - x + 3\end{align}\)

Hence, the required second-degree polynomial is \(P\left( x \right) = {x^2} - x + 3\).