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- Feb 14, 2012

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I've come across a math problem lately and it seems so interesting to me but I don't understand the statement below, which caused me failed to think of a good method to solve it.

"The line is tangent to the graph at exactly two distinct points."

I understand that if we have a function, says, $y=(x^2-4)^2$, then $y=0$ is a tangent line to the curve at two distinct points, namely $(-2,0)$ and $(2,0)$.

But in the problem as stated below, I honestly don't see how could a straight line can be a tangent to the given curve at two distinct points.

Problem:

The line $y=ax+b$ is tangent to the graph of $y=x^4-2x^3-9x^2+2x+8$ at exactly two distinct points. What is the value of $| a+b| $?

The only thing that I could think of to "force" the line $y=ax+b$ be the tangent to the curve is by drawing the green line that touches the curve at its extrema. This is a wrong tangent line, of course because the real tangent line at the extrema has zero slope...I think I am missing something (very important) here...

Any insight that anyone could give would be greatly appreciated.