Let f(x) = max (sin x, cos x) and g(x) = min | vedclass

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(A) Given the quadratic equation $f(x)y^2 + ay + g(x) = 0$ has real roots for all $x \in R$.For a quadratic equation $Ay^2 + By + C = 0$ to have real

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$a \in (-\infty, -\sqrt{2}] \cup [\sqrt{2}, \infty)$

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(A) Given the quadratic equation $f(x)y^2 + ay + g(x) = 0$ has real roots for all $x \in R$.For a quadratic equation $Ay^2 + By + C = 0$ to have real roots,the discriminant $D = B^2 - 4AC \geq 0$.Here,$A = f(x)$,$B = a$,and $C = g(x)$.Thus,$a^2 - 4f(x)g(x) \geq 0$,which implies $a^2 \geq 4f(x)g(x)$.Since $f(x) = \max(\sin x, \cos x)$ and $g(x) = \min(\sin x, \cos x)$,their product $f(x)g(x) = \sin x \cos x$.Therefore,$a^2 \geq 4 \sin x \cos x = 2 \sin(2x)$.For this inequality to hold for all $x \in R$,$a^2$ must be greater than or equal to the maximum value of $2 \sin(2x)$.The maximum value of $2 \sin(2x)$ is $2$.So,$a^2 \geq 2$.This implies $|a| \geq \sqrt{2}$,which gives $a \in (-\infty, -\sqrt{2}] \cup [\sqrt{2}, \infty)$.

Let $f(x) = \max (\sin x, \cos x)$ and $g(x) = \min (\cos x, \sin x)$. If the equation $f(x)y^2 + ay + g(x) = 0$ has real roots for all $x \in R$,then the complete set of values of $a$ is:

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