\[\begin{array}{l}{\text{Определите наименьшее значение параметра }}a{\text{,}}\\{\text{при котором функция }}f\left( x \right) = ax + 7\cos x + 5\sin x\\{\text{возрастает на всей числовой оси}}{\text{.}}\end{array}\]
\[\begin{array}{l}7\sin x - 5\cos x = - \sqrt {74} \cos \left( {x + \arcsin \frac{7}{{\sqrt {74} }}} \right) \Rightarrow \\ - \sqrt {74} \le 7\sin x - 5\cos x \le \sqrt {74} \end{array}\]
\[\begin{array}{l}f'\left( x \right) = a - 7\sin x + 5\cos x = a - \left( {7\sin x - 5\cos x} \right)\\7\sin x - 5\cos x = \sqrt {{7^2} + {5^2}} \left( {\frac{7}{{\sqrt {{7^2} + {5^2}} }}\sin x - \frac{5}{{\sqrt {{7^2} + {5^2}} }}\cos x} \right) = \\\sqrt {74} \left( {\frac{7}{{\sqrt {74} }}\sin x - \frac{5}{{\sqrt {74} }}\cos x} \right) = \\\left[ \begin{array}{l}\frac{7}{{\sqrt {74} }} = \sin \alpha \Rightarrow \frac{5}{{\sqrt {74} }} = \cos \alpha \\\alpha = \arcsin \frac{7}{{\sqrt {74} }}\end{array} \right] = \\\sqrt {74} \left( {\sin x\sin \left( {\arcsin \frac{7}{{\sqrt {74} }}} \right) - \cos x\cos \left( {\arcsin \frac{7}{{\sqrt {74} }}} \right)} \right) = \\ - \sqrt {74} \cos \left( {x + \arcsin \frac{7}{{\sqrt {74} }}} \right)\\f'\left( x \right) = a + \sqrt {74} \cos \left( {x + \arcsin \frac{7}{{\sqrt {74} }}} \right) \Rightarrow \\{\text{при }}a \ge \sqrt {74} {\text{ производная будет неотрицательна }}\forall x.\end{array}\]
\[a = \sqrt {74} \]
