Posted by anandi on Friday, June 3, 2011 at 5:25am.
take the derivative and set that equal to zero
let y = 9tan^2 x + 4cot^2 x
dy/dx = 18tanx(sec^2x) + 8cotx(-csx^2x)
= 0
18(sinx)/(cosx)(1/cos^2x) + 8(cosx/sinx)(-1/sin^2x) = 0
18sinx/cos^3x) - 8cosx/sin^3x = 0
18sinx/cos^3x) = 8cosx/sin^3x
18sin^4x = 8cos^4x
sin^4x/cos^4x = 8/18
tan^4x = 4/9
tanx = ± (4/9)^.25 = ± .8165 appr.
set calculator to radians,
x = .6847 or π-.6847 or π+.6847 or 2π-.6847
sub each of those into original to see which one gives the smaller value
Let z=tan^2(x), z>0
F(z)=9z+4/z,
F'(z)=9-4/z^2, F'(z)=0 if z=2/3
minF(z)=F(2/3)=12
(corresponding value of x exists)
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