Question

The sum of all the local minimum values of the twice differentiable function $f:R\to R$ defined by $f\left(x\right)=x^3-3x^2-\frac{3f^{\prime \prime }\left(2\right)}{2}x+f^{\prime \prime }\left(1\right)$ is:

• A. $-22$
• B. $5$
• C. $-27$
• D. $0$

Answer 1

The correct option is C $-27$

$f^{\prime \prime }\left(x\right)=6x-6$
$f^{\prime \prime }\left(2\right)=6,f^{\prime \prime }\left(1\right)=0$
$f\left(x\right)=x^3-3x^2-9x$
$f^{\prime }\left(x\right)=3x^2-6x-9=3(x+1)(x-3)$
Local min at $x=3$
Local min value $=f\left(3\right)=–27$