If f(x) = sin x - cos x - ax + b is a decreas | vedclass

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Question

(A) Given $f(x) = \sin x - \cos x - ax + b$.For $f(x)$ to be a decreasing function,its derivative $f'(x)$ must be less than or equal to $0$ for all $x

Vedclass · Accepted Answer

$a \ge \sqrt{2}$

Vedclass · Answer

(A) Given $f(x) = \sin x - \cos x - ax + b$.For $f(x)$ to be a decreasing function,its derivative $f'(x)$ must be less than or equal to $0$ for all $x \in R$.$f'(x) = \cos x + \sin x - a$.Since $f(x)$ is decreasing,$f'(x) \le 0 \implies \cos x + \sin x - a \le 0$.This implies $a \ge \sin x + \cos x$ for all $x \in R$.The maximum value of the expression $(\sin x + \cos x)$ is $\sqrt{1^2 + 1^2} = \sqrt{2}$.Therefore,for the inequality $a \ge \sin x + \cos x$ to hold for all $x$,$a$ must be greater than or equal to the maximum value of $(\sin x + \cos x)$.Thus,$a \ge \sqrt{2}$.

If $f(x) = \sin x - \cos x - ax + b$ is a decreasing function for all $x \in R$,then:

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