Question

If $f\left(x\right)=x^2-2x+4$ on $\left[1,5\right]$, then the value of constant $c$ such that $\frac{f\left(5\right)-f\left(1\right)}{5-1}=f\text{'}\left(c\right)$ is

• A. $0$
• B. $1$
• C. $2$
• D. $3$

Answer 1

The correct option is D

$3$

Explanation for the correct option.

Find the value of $c$.

For the function $f\left(x\right)=x^2-2x+4$, the derivative is given as: $f\text{'}\left(x\right)=2x-2$.

Now, the equation $\frac{f\left(5\right)-f\left(1\right)}{5-1}=f\text{'}\left(c\right)$ can be written as:

$$\begin{gathered} \frac{\left(5^2-2\times 5+4\right)-\left(1^2-2\times 1+4\right)}{4}=2c-2 \\ \Rightarrow 25-10+4-1+2-4=8\left(c-1\right) \\ \Rightarrow 16=8\left(c-1\right) \\ \Rightarrow 2=c-1 \\ \Rightarrow 3=c \end{gathered}$$

Hence, the correct option is D.