Question

Let the line $y=100x-199$ be intersect the graph of a function $f(x)=x^3-6x^2+ax-2a+17$, $a\in R$ at three distinct points whose abscissae are $x_1,x_2$ and $x_3\left(x_1<x_2<x_3\right)$ such that $x_3-x_1=6$. If $\underset{x_1}{\overset{x_3}{\int }}(f(x)-98x+198)dx=\lambda$, then find the value of $\left(\frac{\lambda }{2}\right)$

Answer 1

Answer: 9

$f(x)=x^3-6x^2+ax-2a+17$
$f^{\prime }(x)=3x^2-12x+a$
$f^{\prime \prime }(x)=6x-12$
$f^{\prime \prime }(x)=0\Rightarrow x=2$
$f(2)=1$
$(2,1)$ is the point of inflection on the graph $y=f(x)$ Straight line $y=100x-199$ is also passing through $(2,1)$
$x_1+x_3=4\text{ and }{x}_3-x_1=6$

$\underset{x_1}{\overset{x_2}{\int }}(f(x)-98x+198)dx$
$=\underset{x_1}{\overset{x_3}{\int }}(f(x)-100x+199+2x-1)dx=\underset{x_1}{\overset{x_3}{\int }}(2x-1)dx$
$={\left(x^2-x\right)}_{x_1}^{x_3}=x_3^2-x_3-\left(x_1^2-x_1\right)$
$=\left(x_3-x_1\right)\left(x_3+x_1\right)-\left(x_3-x_1\right)$
$=4\times 6-6=18$
$\therefore \frac{\lambda }{2}=9$