find minimum or maximum of f(x)=x^2+9x-36

Question

find the minimum or maximum of this problem?

Answer 1

[Using calculus: f'(x)=2x+9=0 at minimum, so x=-4.5, f(-4.5)=-56.25. The minimum (parabola vertex) at (-4.5,-56.25).]

SOLUTION WITHOUT USING CALCULUS

f(x)=x²+9x-36=(x+12)(x-3). The minimum (the vertex) of parabola f(x) is halfway between the roots 3 and -12, that is, ½(3-12)=-9/2=-4.5. The sign of x² is positive, indicating an upright parabola (arms extending upward), so the vertex is the lowest point, a minimum.