Question

Y=-4/x [-1/2;-1/4] найти наибольшее и наименьшее значения функции

Answer 1

Yнаиб=-4/-1/4=16;унаим=-4/-1/2=8

Answer 2

To find the maximum and minimum values of the function $Y = -\frac{4}{x}$ within the interval $[-1/2, -1/4]$, we need to evaluate the function at the endpoints of the interval and at critical points (if any) within the interval. Critical points are points where the derivative of the function is either zero or undefined.

Step 1: Evaluate the function at the endpoints of the interval: $x = -1/2$: $Y = -\frac{4}{-1/2} = 8$

$x = -1/4$: $Y = -\frac{4}{-1/4} = -16$

Step 2: Find the critical points: To find the critical points, we need to find where the derivative of the function is zero or undefined. The derivative of $Y = -\frac{4}{x}$ with respect to $x$ can be found using the quotient rule:

$Y' = \frac{d}{dx} \left(-\frac{4}{x}\right) = \frac{-\frac{d}{dx}(4) \cdot x - 4 \cdot \frac{d}{dx}(x)}{x^2} = \frac{4}{x^2}$

Now, we set the derivative equal to zero to find critical points:

$\frac{4}{x^2} = 0$

Since the denominator $x^2$ can never be zero, there are no critical points within the interval $[-1/2, -1/4]$.

Step 3: Analyze the results: The maximum and minimum values occur either at the endpoints or at critical points. Since there are no critical points within the interval, we only need to compare the function values at the endpoints.

$Y = 8$ at $x = -1/2$

$Y = -16$ at $x = -1/4$

So, the maximum value of the function within the interval is $Y = 8$ (at $x = -1/2$), and the minimum value is $Y = -16$ (at $x = -1/4$).