Question

If Rolle's theorem holds for the function $f\left(x\right) = 2x^{3} + bx^{2} + cx, x \epsilon \left[-1, 1\right],$ at the point $x = \frac{1}{2},$ then $2b + c$ equals :

• A. 1
• B. -1
• C. 2
• D. -3

Answer 1

Answer: (B)

$f(-1)=f(1)$ and $f'\left(\frac{1}{2}\right)=0$
$\Rightarrow -2+b-c=2+b+c \Rightarrow c=-2$
and $f'(x)=6 x^{2}+2 b x +c \Rightarrow f'\left(\frac{1}{2}\right)=\frac{3}{2}+b-2=0$
$\Rightarrow b=\frac{1}{2} \Rightarrow 2 b +c=-1$