Question

Value of $c$ in Lagrange's mean value theorem for the function $f\left(x\right)=x^2-3x+5$ in the interval $[1,4]$ is:

• A. $2.5$
• B. $3.5$
• C. $3$
• D. $2$

Answer 1

The correct option is A $2.5$

The given quadratic function is continuous and differentiable on the entire set of real numbers. Hence, we can apply Lagrange's mean value theorem.
$f^{\prime }\left(x\right)=2x-3$
$f^{\prime }\left(c\right)=\frac{f\left(b\right)-f\left(a\right)}{b-a}$,
$\Rightarrow 2c-3=\frac{(4^2-3\cdot 4+5)-(1^2-3\cdot 1+5)}{4-1}$
$\Rightarrow 2c-3=2\Rightarrow 2c=5\Rightarrow c=2.5$