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shubhamsrg

  • 2 years ago

how to prove : sin^3 x + cos ^3 x <= 1

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  1. NewbieCarrot
    • 2 years ago
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    Did you learn integration before?

  2. shubhamsrg
    • 2 years ago
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    yes..

  3. shubhamsrg
    • 2 years ago
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    ??

  4. NewbieCarrot
    • 2 years ago
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    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

  5. shubhamsrg
    • 2 years ago
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    didnt get you..please elaborate ..

  6. NewbieCarrot
    • 2 years ago
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    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).

  7. shubhamsrg
    • 2 years ago
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    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..

  8. shubhamsrg
    • 2 years ago
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    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 !!!!!!!!!!!!!!!!!

  9. shubhamsrg
    • 2 years ago
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    thanks,,@NewbieCarrot

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