Question

The graphs $y=2x^3-4x+2$ and $y=x^3+2x-1$ intersect at exactly three distinct points. The slope of the line passing through two of these points

• A. is equal to $4$
• B. is equal to $6$
• C. is equal to $8$
• D. is not unique

Answer 1

The correct option is C is equal to $8$

Let $(x_1,y_1)$ and $(x_2,y_2)$ be two of these points.
Given $y=2x^3-4x+2$ and $y=x^3+2x-1$
$\therefore y_1=2x_1^3-4x_1+2\cdots \left(1\right)$
and $y_1=x_1^3+2x_1-1\cdots \left(2\right)$
Equation$\left(2\right)\times 2-$ Equation$\left(1\right)$, we get
$y_1=8x_1-4\cdots \left(3\right)$

Simillarly, $y_2=2x_2^3-4x_2+2\cdots \left(4\right)$
and $y_2=x_2^3+2x_2-1\cdots \left(5\right)$
Equation$\left(5\right)\times 2-$ Equation$\left(4\right)$, we get
$y_2=8x_2-4\cdots \left(6\right)$

Equation$\left(6\right)-$ Equation$\left(3\right)$, we get
$y_2-y_1=8x_2-4-8x_1+4$
$\Rightarrow y_2-y_1=8(x_2-x_1)$
$\Rightarrow \frac{y_2-y_1}{x_2-x_1}=8$