Show that 4n + 3^n > 6n, for all integers, n > 2.

Induction.

Step-by-step explanation:

I will prove it with induction. For n=3:

Suppose that the inequality is true for a k>3, that is to say

Then,

because and because k>0.

Then,

.

In conclusion, for all integers n> 2 the inequality is true.

solution:

the given function is ,

      =

the three roots of this quadratic function is , 0 ,8 and 10.all the roots are real and rational.

if a quadratic equation is of the form ,

sum of the roots = a + b +c= 0+8+10=18=  

product of roots taken two at a time = a b + b c+ c a = 0+80+0=80 =  

product of roots = a b c = 0=  

there are 11 doors in a hall starting from 0 to 11.the entry in the hall is permitted through gate number 0 and exit is allowed through only gate number 8 and 10.so, three roots(0,8,10) of quadratic function is given by entry and exit route of this hall.