If sin^4 alpha + 4 cos^4 beta + 2 = 4sqrt{2} | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) Using the $AM \geq GM$ inequality for the four terms $\sin^4 \alpha, 4 \cos^4 \beta, 1, 1$:$\frac{\sin^4 \alpha + 4 \cos^4 \beta + 1 + 1}{4} \geq

Vedclass · Accepted Answer

$-\sqrt{2}$

Vedclass · Answer

(D) Using the $AM \geq GM$ inequality for the four terms $\sin^4 \alpha, 4 \cos^4 \beta, 1, 1$:$\frac{\sin^4 \alpha + 4 \cos^4 \beta + 1 + 1}{4} \geq (\sin^4 \alpha \cdot 4 \cos^4 \beta \cdot 1 \cdot 1)^{1/4}$$\frac{\sin^4 \alpha + 4 \cos^4 \beta + 2}{4} \geq (4 \sin^4 \alpha \cos^4 \beta)^{1/4} = \sqrt{2} \sin \alpha \cos \beta$$\sin^4 \alpha + 4 \cos^4 \beta + 2 \geq 4\sqrt{2} \sin \alpha \cos \beta$Given the equality holds,$AM = GM$,so $\sin^4 \alpha = 4 \cos^4 \beta = 1$.Thus,$\sin^4 \alpha = 1 \Rightarrow \sin \alpha = 1$ (since $\alpha \in [0, \pi]$).And $4 \cos^4 \beta = 1$ $\Rightarrow \cos^2 \beta = 1/2$ $\Rightarrow \cos \beta = \pm 1/\sqrt{2}$.Since $\beta \in [0, \pi]$,$\sin \beta = 1/\sqrt{2}$.If $\cos \beta = 1/\sqrt{2}$,then $\alpha = \pi/2, \beta = \pi/4 \Rightarrow \cos(\alpha + \beta) = \cos(3\pi/4) = -1/\sqrt{2}$.If $\cos \beta = -1/\sqrt{2}$,then $\alpha = \pi/2, \beta = 3\pi/4 \Rightarrow \cos(\alpha + \beta) = \cos(5\pi/4) = -1/\sqrt{2}$.

If $\sin^4 \alpha + 4 \cos^4 \beta + 2 = 4\sqrt{2} \sin \alpha \cos \beta$ and $\alpha, \beta \in [0, \pi],$ then $\cos(\alpha + \beta)$ is equal to

Similar Questions

If $0 < \theta < \frac{\pi}{2}$ and $\tan 3 \theta \neq 0$,then $\tan \theta + \tan 2 \theta + \tan 3 \theta = 0$ if $\tan \theta \cdot \tan 2 \theta = k$,where $k =$

If $\cos \left(\frac{\alpha-\beta}{2}\right)=2 \cos \left(\frac{\alpha+\beta}{2}\right)$,then $\tan \frac{\alpha}{2} \tan \frac{\beta}{2}=$

The number of real roots of the equation $e^{\sin x} - e^{-\sin x} - 4 = 0$ is:

The value of the expression $\frac{1+\sin 2 \alpha}{\cos (2 \alpha-2 \pi) \tan \left(\alpha-\frac{3 \pi}{4}\right)} - \frac{1}{4} \sin 2 \alpha \left[\cot \frac{\alpha}{2}+\cot \left(\frac{3 \pi}{2}+\frac{\alpha}{2}\right)\right]$ is:

If $A$ and $B$ are acute angles such that $\sin A = \sin^2 B$ and $2 \cos^2 A = 3 \cos^2 B$,then $(A, B) =$

Vedclass Test Series

Exam Paper Generator

Online Exam Module