What is the maximum value of k so

evighetl2b

evighetl2b

Answered question

2022-03-17

What is the maximum value of k so that x2+4(sin2{x}tan2{x}+cos2{x}cot2{x}+k2xsec{x}csc{x})=0 has real roots.

Answer & Explanation

clugiarh0j

clugiarh0j

Beginner2022-03-18Added 7 answers

x2(4sec{x}csc{x})x+4(sin2{x}tan2{x}+cos2{x}cot2{x}+k2)=0
Look for the discriminant, for real roots D0
16sec2xcsc2x16(sin2xtan2x+cos2xcot2x+k2)0
1sin2xcos2xsin4xcos2xcos4xsin2xk20
1sin6xcos6xsin2xcos2xk2
sin2x+cos2xsin6xcos6xsin2xcos2xk2
sin2x(1sin4x)+cos2x(1cos4x)sin2xcos2xk2
sin2x(1sin2x)(1+sin2x)+cos2x(1cos2x)(1+cos2x)sin2xcos2xk2
sin2xcos2x(1+sin2x+1+cos2x)sin2xcos2xk2
3k23k3 So maximum value of k is 3

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