Function question 2 for powerplay

for x ≤ -1 , the inequality is not satisfied...!!!

xεR
R>0

if not then correct me....

no one has given a solution though

actually prophet this may be the soln

every x^odd function with no constants can be represented as

let the horizontal axis be x and the vertical be y

sankara does that mean that every odd degree polynomial can have one real root?

so many others have given x>0. Is that something you have encountered before or do you guys know how to do it?

x¹¹-x⁸+x⁴-x²+x>0
it is easy to see x=0 is not a solution
so dividing throughout with x
x¹⁰-x⁷+x⁴-x > -1
x⁷(x³-1) + x(x³-1) > -1
(x⁷+x)(x³-1) > -1
so it is easy to see x>0
pls correct me if wrong

when you divide by x and you preserve the direction of inequality you are already presuming that x>0. You are on the right track, but you will have to tweak the solution a bit.

My solution:

If x≤0, its obvious that the condition is not satisfied.

For proving for x>0, we take two cases 0<x<1 and x>1.

For case 1, write the expression as x¹¹+(x⁵-x⁸)+(x-x²). Each of the bracketed expressions is non-negative and x¹¹>0, so the expression is positive

For case 2, write it as (x¹¹-x⁸)+(x⁵-x²)+x and by the same reasoning, the expression is positive for this case too.