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

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(A) Given $h(y) = f(x)y^2 + ay + g(x) = 0$ has real roots for all $x \in R$.This implies the discriminant $D \geq 0$.$D = a^2 - 4f(x)g(x) \geq 0 \Righ

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

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(A) Given $h(y) = f(x)y^2 + ay + g(x) = 0$ has real roots for all $x \in R$.This implies the discriminant $D \geq 0$.$D = a^2 - 4f(x)g(x) \geq 0 \Rightarrow a^2 \geq 4f(x)g(x)$.Since $f(x) = \max(\sin x, \cos x)$ and $g(x) = \min(\sin x, \cos x)$,their product is $f(x)g(x) = \sin x \cos x$.Thus,$a^2 \geq 4 \sin x \cos x = 2 \sin(2x)$.For this 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 $2$.Therefore,$a^2 \geq 2$,which implies $|a| \geq \sqrt{2}$.This gives $a \in (-\infty, -\sqrt{2}] \cup [\sqrt{2}, \infty)$.

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

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