341funnily, none of your questions are applications of derivatives (only the 1st one but not by much)
Ques1) Show that the extremum value of the function
f(x) = 1 / (sinx+4) - 1 / (cos x -4), where x belongs to R is
2√2 / (4 √2 + 1).
Ques2) If a2x4 + b2y4 = c6, then show that the max value of xy is
c3 / √2ab.
Ques3) If x7 - x5+x3-x=3a , x>0, then show that x8 will bw greater than 1 + 6a.
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3 Answers
341last one is nice, but it appears to be true for x>1 (not x>0) as you have typed
from the given equation we get x^7-x^5 + x^3-x = 3a \Rightarrow (x^2-1)(x^5+x) = 3a \Rightarrow x^2-1 = \frac{3a}{x^5+x}
Now we have to prove that x8>1+6a or equivalently x8-1>6a
x^8 -1 = (x^2-1)(x^6+x^4+x^2+1) = \frac{3a}{x^5+x}(x^6+x^4+x^2+1)
Hence it suffices to prove that \frac {x^6+x^4+x^2+1}{x^5+x}> 2 or x^6+x^4+x^2+1 > 2x^5+2x
which is true by AM-GM as x6+x4>2x5 and x2+1>2x (strict ineq because x>1)
or could write x^6-2x^5 + x^4+x^2-2x+1 = (x^3-x^2)^2 + (x-1)^2 >0to prove the same thing.
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