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Consider the function

f(x)= ax^2 if x>a

f(x)= ax+2a if x

where a is some number.

What is the largest value of a such that the graph of y= f(x) intersects every horizontal line at least once?

 Jan 5, 2021
edited by hihihi  Jan 5, 2021
edited by hihihi  Jan 5, 2021
edited by hihihi  Jan 5, 2021
edited by hihihi  Jan 5, 2021
 #1
avatar+303 
0

For x < a, the graph of y = f(x) is the same as the graph of y = ax+2a, which is a line with slope a and which passes through the point (a, a^2+2a). For \(x \ge a\), the graph of y = f(x) is the same as the graph of y = ax^2, which is a parabola passing through the point (a, a^3). Notice that the parabola only ever takes nonnegative values. Therefore, the line portion of the graph must have positive slope, because it must intersect horizontal lines which lie below the $x-$axis. Thus, $a > 0.$ For $a > 0,$ the line portion of the graph passes through all horizontal lines with height less than or equal to $a^2+2a,$ and the parabola portion of the graph passes through all horizontal lines with height greater than or equal to $a^3.$ Therefore, all horizontal lines are covered if and only if \(a^2 + 2a \ge a^3.\) Since  a > 0, we can divide by a to get a\( + 2 \ge a^2\), so \(0 \ge a^2 - a - 2 = (a-2) ( a+1)\). This means that \(-1 \le a \le 2,\) so the greatest possible value of a is \(\boxed{2}\)

 Jan 5, 2021

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