If A=sin2x +cos4x,then for all real x:
a.34≤A≤1316
b.34≤A≤1
c.1316≤A≤1
d.1≤A≤2
please provide a detailed solution to the above problem.
Thank You.
106as 0 ≤ cos2x ≤ 1
=> cos4x ≤ cos2x
=> sin2x + cos4x ≤ sin2x + cos2x = 1 ... (1)
Let cos2x = t
then sin2x + cos4x = t2 - t + 1 and its min value occurs at t = 1/2 (which lies in domain) and its min. value is 3/4
So, 3/4 ≤ sin2x + cos4x ≤ 1
1708$\textbf{Let $\mathbf{A=sin^2\;x+cos^4\;x=sin^2x+cos^2x.\left(1-sin^2\;x\right)$}}$\\\\\\ $\textbf{So $\mathbf{A=1-cos^2\;x.sin^2\;x=1-\frac{1}{4}.\left(2sinx.cosx\right)^2}$}$\\\\\\ \mathbf{A=1-\frac{1}{4}.sin^2\;2x\Leftrightarrow {\frac{1}{4}sin^2\2x=1-A.....................................(1)}}$\\\\\\ \textbf{Now $\mathbf{0 \leq sin^2\;2x\leq 1\Leftrightarrow 0\leq \frac{1}{4}.sin^2\;2x\leq \frac{1}{4}}$}$\\\\\\ $\textbf{Now Put value from equation..(1)}$\\\\ \mathbf{0\leq 1-A\leq \frac{1}{4}\Leftrightarrow -\frac{1}{4}\leq A-1\leq 0}$\\\\ \textbf{So $\boxed{\boxed{\mathbf{\frac{3}{4}\leq A\leq 1}}}$}$
1Diid u all notice this question was a FAMOUS JEE question gosh AIEE standard is too low ... suprisingly my and ashishs bhaiya solution was exactly same ( i did it a month back) true great minds think alike :)
31 - cos2x + cos4x
or, cos4x - 2/2*cos2x + 1/4 - 1/4 + 1
or, (cos2x - 1/2)2 + 3/4
so the maximum value is at cos2x = 1
that is 1/4 + 3/4 = 1
the minimum value is at cos2x = 1/2
that is 3/4.
hence
3/4≤A≤1