2010 AMC 12A Problem 21

Problem 21 of 25HarderAlgebra

The graph of

y=x^6-10x^5

+29x^4-4x^3+ax^2

lies above the line

y=bx+c

except at three values of

x,

where the graph and the line intersect. What is the largest of those values?

Show solution

Solution

Let

f(x)

be the graph minus the line. It is nonnegative and vanishes at three points, each a double root, so

f(x)=\big(x^3-Ax^2+Bx-C\big)^2.

Matching coefficients gives

-2A=-10\Rightarrow A=5,

then

A^2+2B=29\Rightarrow B=2,

then

-2C-2AB=-4\Rightarrow C=-8.

Thus the cubic is

x^3-5x^2+2x+8

=(x+1)(x-2)(x-4),

with roots

-1,2,

and

4.

The largest is

4.

Thus, A is the correct answer.