What is the smallest real number

Question

What is the smallest real number a for which the function f(x) = 4x² − 12x − 5 + 2a will always be nonnegative for all real numbers x?

Answer 1

f(x) = 4x^2 -12x -5 +2a

is an upward opening parabola with vertex = the minimum point. choose a so that the vertex is on the x axis with y coordinate = 0

rewrite the equation in vertex form

f(x) = 4(x^2-3x ) -5+2a

f(x) =4(x^2-3x +(3/2)^2) -5 +2a-4(3/2)^2

f(x) = 4(x-3/2)^2 -5+2a -9

f(x) =4(x-3/2)^2 +2a-14

the y coordinate of the vetex = 2a-14

set it equal to zero and solve for a

2a-14=0

a =14/2 = 7