Mixed problems of inequality and function~ If the function f (x) = TX ^ 2 - (22T + 60) x + 144t (x > 0) (1) to make f (x) ≥ 0 hold, find the minimum value of T; (2) to make f (x) = 0, find the value range of X to make t > 20 hold

Mixed problems of inequality and function~ If the function f (x) = TX ^ 2 - (22T + 60) x + 144t (x > 0) (1) to make f (x) ≥ 0 hold, find the minimum value of T; (2) to make f (x) = 0, find the value range of X to make t > 20 hold

(1) F (x) = TX ^ 2 - (22T + 60) x + 144t ≥ 0
tX^2-22tx-60x+144t ≥0
t(x^2-22x+144) ≥60x,x^2-22x+144=(x-11)^2+23＞0
T ≥ 60x / (x ^ 2-22x + 144)
The numerator denominator of 60x / (x ^ 2-22x + 144) is divided by X, and the denominator is based on the basic inequality. The maximum value is 30
So t ≥ 30, Tmin = 30
(2) The same as above, t (x ^ 2-22x + 144) = 60x
The separation parameter is 60x / (x ^ 2-22x + 144) = t > 20
It is reduced to x ^ 2-25x + 144 < 0, so 9 < x < 16