Find the largest value of c such that -2 is in the range of \(f(x)=x^2+3x+c\) .

mathtoo
Aug 27, 2018

#1**+3 **

OK...similar to the last one

This parabola also turns upward....

The x coordinate of the vertex is -3/2

So...put this into the function and set up an inequality where the function is greater than -2

Thus...when we find solutions....we wil exclude those from the possible answers...so we have

(-3/2)^2 + 3(-3/2) + c > -2 simplify

9/4 - 9/2 + c > -2

-9/4 + 2 > - c

-9/4 + 8/4 > - c

-1/4 > - c divide both sides by -1 and reverse the inequality sign

1/4 < c

So...-2 will be in range whenever c ≤ 1/4

See the graph, here : https://www.desmos.com/calculator/8uqpiabiz6

Note that when c = 1/4....-2 is still in the range

But..for instance..when c = 1/2, -2 is out of range

CPhill
Aug 27, 2018