If the Rolle's theorem holds for the function f(x)=2x^3+ax^2+bx in the interval [-1,1] for…

Question

If the Rolle's theorem holds for the function $f(x)=2x^3+ax^2+bx$ in the interval $[-1,1]$ for the point $c=\dfrac12$, then the value of $2a+b$ is:

Accepted Answer

$f(1)=f(-1)$: $2+a+b=-2+a-b\Rightarrow b=-2$.

$f'\left(\dfrac12\right)=0$: $\dfrac32+a+b=0\Rightarrow a=2-\dfrac32=\dfrac12$.

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

Solutions · 1