Consider the function
f(x)= ax^2 + a if x > a
f(x)= ax + 4a if x <= a
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?
if x>=a, the graph y=f(x) is the same graph as y=ax^2
line in the graph has positive slope, since the parobola only has nonegative values
so, a>0, and thus, the line in the graph passes through every single horizantal line <= a^2+2a
the parabola region passes through every single horizantal line >=a^3
a+2>=a^2 (since a>0 and can divide by a)
greatest value of a is 2