if x is real, the maxium value is?

Question

if x is real, the maxium value of          3x² + 9x + 17       is

                                                                  3x² - 9x  + 7   

Answer 1

First, note that 3x²-9x+7 has only complex zeroes, so there is no vertical asymptote.

Using calculus the expression needs to be differentiated:

[(3x²-9x+7)(6x+9)-(3x²+9x+17)(6x-9)]/(3x²-9x+7)²=

[18x³-27x²-39x+63-(18x³+27x²+21x-153)]/(3x²-9x+7)²=

(-54x²-60x+216)/(3x²-9x+7)²=0 at max or min.

-54x²-60x+216=0, 9x²+10x-36=0,

x=(-10±√(100+1296))/18=(-10±√1396)/18=(-5±√349)/9, x=1.5202 or -2.6313.

x=-2.6313 is minimum, and x=1.5202 is maximum.

When x=-2.6313 the expression is 0.2738 approx.

When x=-2, the expression is (12-18+17)/(12+18+7)=11/37=0.2973 (approx)>0.2738.

When x=-3, the expression is (27-27+17)/(27+27+7)=17/61=0.2787 (approx)>0.2738, making x=-2.6313 the minimum.

When x=1.5202 the expression is 149.7262 approx.

When x=1 the expression is 29. When x=2 the expression is 47.

When x=1.5202 the expression is 149.7262, much greater than either 29 or 47, making 149.7262 (approx) the maximum.