Question

Максималное значение функции 2sinx+7cosx

Answer 1

Решение смотрите на фотке.

Answer 2

To find the maximum value of the function $f(x) = 2\sin(x) + 7\cos(x)$, we need to analyze its behavior. The maximum value of $f(x)$ occurs when the sum $2\sin(x) + 7\cos(x)$ is at its maximum value.

The maximum value of the sum of two trigonometric functions $a\sin(x) + b\cos(x)$ can be found using the following formula:

$R = \sqrt{a^2 + b^2}$

Where $R$ represents the amplitude of the sum. In our case, $a = 2$ and $b = 7$, so:

$R = \sqrt{2^2 + 7^2} = \sqrt{4 + 49} = \sqrt{53}$

Now, to find the maximum value of $f(x)$, we need to determine the phase shift (horizontal shift) that occurs with the sum $2\sin(x) + 7\cos(x)$.

The general form of the sum of two trigonometric functions with different phase shifts is:

$a\sin(x + \alpha) + b\cos(x + \beta)$

where $\alpha$ and $\beta$ represent the phase shifts.

For our function $2\sin(x) + 7\cos(x)$, there's no phase shift ($\alpha = \beta = 0$), so the maximum value is simply equal to $R$:

$f_{\text{max}} = \sqrt{53}$

Thus, the maximum value of the function $2\sin(x) + 7\cos(x)$ is $\sqrt{53}$ and it occurs when $x$ produces the maximum value of $f(x)$.