Question

If $f\left(x\right)=mx+c$ and if $f\left(0\right)=f\text{'}\left(0\right)=1$ then $f\left(2\right)=$

• A. $1$
• B. $2$
• C. $3$
• D. $-3$

Answer 1

The correct option is C

$3$

Explanation for the correct option:

Step 1: Finding the value of $m$ and $c$:

Given that,

$f\left(x\right)=mx+c$

Substitute $x=0$ in the given equation,

$$\begin{gathered} f\left(0\right)=\left(0\right)+c \\ \Rightarrow c=1\left[\because f\left(0\right)=1\right] \end{gathered}$$

Now, differentiate the given equation with respect $x$, we get,

$$\begin{gathered} f\text{'}\left(x\right)=m \\ \Rightarrow f\text{'}\left(0\right)=m \\ \Rightarrow m=1 \end{gathered}$$

Step 2: Finding the value of $f\left(2\right)$:

Substitute the value of $m$ and $c$ in the given equation,

$f\left(x\right)=x+1$

Now, Substitute $x=2$ in the above equation.

$\begin{array}{rcl}f\left(2\right)& =& 2+1\\ & =& 3\end{array}$

Hence, the correct option is C.