Question

Find all the polynomials of degree$\mathbf{\le }\mathbf{2}$[a polynomial of the form$\mathbf{}\mathbf{}\mathbf{f}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\mathbf{a}\mathbf{+}\mathbf{bt}\mathbf{+}{\mathbf{ct}}^{\mathbf{2}}$] whose graph goes through the points $\mathbf{}\mathbf{(}\mathbf{1}\mathbf{,}\mathbf{3}\mathbf{)}\mathbf{and}\mathbf{(}\mathbf{2}\mathbf{,}\mathbf{6}\mathbf{)}\mathbf{,}$such that $\mathbf{f}\mathbf{\text{'}}\mathbf{\left(}\mathbf{1}\mathbf{\right)}\mathbf{=}\mathbf{1}$[where$\mathit{f}\mathbf{\text{'}}\mathbf{\left(}\mathit{t}\mathbf{\right)}$denotes the derivative].

Short answer

The polynomial of degree $\le 2$[a polynomial of the form $\mathrm{f}\left(\mathrm{t}\right)=\mathrm{a}+\mathrm{bt}+{\mathrm{ct}}^2$] whose graph goes through the points $(1,3)\mathrm{and}(2,6)$such that $\mathrm{f}\text{'}\left(1\right)=1$is$f\left(\mathrm{t}\right)=4-3\mathrm{t}+2{\mathrm{t}}^2$ .

Step-by-step solution

Consider the points and substitute these in the standard equation

A polynomial of degree 2 is of the form$\mathit{f}\mathbf{\left(}\mathit{t}\mathbf{\right)}\mathbf{=}\mathit{a}\mathbf{+}\mathit{b}\mathit{t}\mathbf{+}\mathit{c}{\mathit{t}}^{\mathbf{2}}$. Consider a polynomial of degree 2 and substitute given point in them as:

role="math" localid="1659347379302" $$\begin{gathered} f_1\left(t\right)=a+b{t}_1+c{t}_1^2 \\ 3=a+b\left(1\right)+a{\left(1\right)}^2 \end{gathered}$$

$$\begin{gathered} f_2\left(t\right)=a+b{t}_2+c{t}_2^2 \\ 6=a+b\left(2\right)+a{\left(2\right)}^2 \end{gathered}$$

Consider the derivative of the polynomial $\mathrm{f}\left(\mathrm{t}\right)=\mathrm{a}+\mathrm{bt}+{\mathrm{ct}}^2\mathrm{as}\mathrm{f}\text{'}\left(1\right)=1:$

$$\begin{gathered} f\text{'}\left(t\right)=b+2ct \\ f\text{'}\left(1\right)=b+2c\left(1\right) \\ 1=b+2c \end{gathered}$$

Rearrange the terms of the above equations

Consider the simplified equations.

$$\begin{gathered} 3=a+b+c.......\left(1\right) \\ 6=a+2b+4c........\left(2\right) \\ 1=0+b+2c.....\left(3\right) \end{gathered}$$

Solve the above equations (1), (2) and (3)

Represent the above obtained equations in terms of matrix.

$\left(\begin{array}{ccc}1& 1& 1\\ 1& 2& 4\\ 0& 1& 2\end{array}\right)\left(\begin{array}{c}a\\ b\\ c\end{array}\right)=\left(\begin{array}{c}3\\ 6\\ 1\end{array}\right)$

Upon solving the values of a,b and c are obtained as$a=4,b=-3,c=2$

Substitute these values in the standard equation of the $\le 2$degree polynomial.

$\begin{array}{l}f\left(t\right)=\left(4\right)+(-3)t+\left(2\right)t^2\\ f\left(t\right)=4-3t+2t^2\end{array}$

The polynomial of degree $\le 2$[a polynomial of the form $f\left(t\right)=a+bt+c{t}^2$] whose graph goes through the points $(1,3)and(2,6)$such that $\mathrm{f}\text{'}\left(1\right)=1\mathrm{is}\mathrm{f}\left(\mathrm{t}\right)=4-3\mathrm{t}+2{\mathrm{t}}^2.$