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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?

 Jun 12, 2021
 #1
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+1

if x

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+2a>=a^3

a+2>=a^2 (since a>0 and can divide by a)

0>a^2-a-2

0>(a-2)(a+1)

-1<=a<=2

greatest value of a is 2

 

JP

 Jun 12, 2021

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