If Rolle's theorem holds for the function f(x)=2x^3+bx^2+cx, x[-1,1], at the point x=12, t…

Question

If Rolle's theorem holds for the function $f(x)=2x^3+bx^2+cx$, $x\in[-1,1]$, at the point $x=\dfrac12$, then $2b+c$ equals:

Accepted Answer

Rolle's condition $f(1)=f(-1)$: $2+b+c=-2+b-c\Rightarrow c=-2$.

$f'(x)=6x^2+2bx+c$; $f'\left(\dfrac12\right)=0$: $\dfrac32+b-2=0\Rightarrow b=\dfrac12$.

$2b+c=1-2=-1$.

Solutions · 1