Question

Let the line $y=100 x-199$ be intersect the graph of a function $f(x)=x^3-6 x^2+a x-2 a+17$, $a \in R$ at three distinct points whose abscissae are $x_1, x_2$ and $x_3\left(x_1

Answer 1

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

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