A. maths Inequalities (Ergent HW!!!!!!!!!!!!!!) 10pts

For the expression be non- negative for all real values of x,

x^2-(k-4)x+k^2-5k+4≧0, then

△≦0

[-(k-4)]^2-4(1)(k^2-5k+4)≦0

(k-4)^2-4(k-4)(k-1)≦0

(k-4)[(k-4)-4(k-1)]≦0

(k-4)(k-4-4k+4)≦0

(k-4)(-3k)≦0

(k-4)(3k)≧0

So k ≦ 0 or k ≧ 4

∵x^2-(k-4)x+k^2-5k+4 is non-negative for all real values of x
∴x^2-(k-4)x+k^2-5k+4≧0
x^2-(k-4)x+(k-4)(k-1)≧0
delta≦0
[-(k-4)]^2-4(k-4)(k-1)≦0
-3k(k-4)≦0
k(k-4)≧0
k≧4 or k≦0