Show that (I)2#x^2#-8x+11>0 ∀ x (ii)6x<9#x^2#+5 ∀ x. ?

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Answer 1 · 2018-07-02T14:58:12

Please see the explanation below

First part #(1)#

The equation is

#f(x)=2x^2-8x+11#

Calculate the discriminant

#Delta=(-8)^2-4(2)(11)#

#=64-88#

#=-24#

As the discriminant is #Delta<0#, the function is

#f(x)<0#, #AA x in RR#

graph{2x^2-8x+11 [-9.57, 15.74, -2.05, 10.61]}

Second part #(1)#

The equation is

#6x<9x^2+5#

#=>#, #9x^2-6x+5>0# , this has to be proved

So, calculate the discriminant

#Delta=(-6)^2-4(9)(5)#

#=36-180#

#=-144#

As the discriminant is #Delta<0#

Therefore,

#9x^2-6x+5>0#

#=>#, #9x^2+5>6x#

#=>#, #6x<9x^2+5#

graph{9x^2-6x+5 [-5.86, 8.184, 0.826, 7.846]}