Question

The value of $c$ in the Lagrange's mean value theorem for the function $f \left(x\right)=x^{3}-4x^{2}+8x+11,$ when $x\,\epsilon\,\left[0, 1\right]$

• A. $\frac{4-\sqrt{7}}{3}$
• B. $\frac{2}{3}$
• C. $\frac{\sqrt{7}-2}{3}$
• D. $\frac{4-\sqrt{5}}{3}$

Answer 1

Answer: (A)

$f \left(0\right)=11$
$f \left(1\right)=16$
$\frac{f \left(1\right)-f \left(0\right)}{1-0}=3c^{2}-8c+8$
$\Rightarrow 3c^{2} - 8c + 8 = 5$
$\Rightarrow 3c^{2} - 8c + 3 = 0
c\,\in\,\left[0, 1\right] \Rightarrow c=\frac{4-\sqrt{7}}{3}$