Question

how to prove :

sin^3 x + cos ^3 x <= 1

Answer 1

Did you learn integration before?

Answer 2

yes..

Answer 3

??

Answer 4

Use integration to find the critical points and take that back to your equation sin^3(x)+cos^3(x). But I think it should be not less than 1

Answer 5

didnt get you..please elaborate ..

Answer 6

let f(x)=sin^3(x)+cos^3(x), then f'(x)=3sin^2(x)cos(x)-3cos^2(x)sin(x) and solve f'(x)=0 for x and use x to find the maximum or minimum of f(x).

Answer 7

fair enough i guess..

i got x=0,pi/2 and pi/4

0 and pi/2 yield f(x) =1
and the other one gives 1/sqrt(2)

seems correct !! hmm..

Answer 8

in the mean time,,i came across this :

we know

sin^2x ( sinx -1) <= cos^2x ( 1- cosx)

equality will hold for x=0 or pi/2

so,
sin^3 x - sin^2 x <= cos^2x - cos^3x

=> sin^3x + cos^3 c <= 1

!!!!!!!!!!!!!!!!!

Answer 9

thanks,,@NewbieCarrot