Question

Определите значение х при котором функция у=-х²+2х-1 принимает наибольшее значение найдите это значение

Answer 1

График функции у=-х²+2х-1 - парабола, ветви которой направлены вниз,
т.к. а=-1 <0, поэтому наибольшее значение функции - это ордината вершины параболы.
Находим координаты вершины параболы:
х(в)=-2/2(-1)=-2/-2=1
у(в)=-1²+2*1+1=-1+3=2   -  ордината вершины
Значит, у(наиб.)=2

Answer 2

Finding the Maximum Value of the Function

To find the value of x at which the function f(x) = -x² + 2x - 1 takes its maximum value, we can use calculus. The maximum value occurs at the vertex of the parabola represented by the function.

The vertex of a parabola in the form f(x) = ax² + bx + c is given by the x-coordinate x = -b/2a. In this case, a = -1, b = 2, and c = -1.

Using the formula, we can calculate the x-coordinate of the vertex:

x = -b/2a = -2/(2*(-1)) = -2/(-2) = 1.

Therefore, the value of x at which the function f(x) = -x² + 2x - 1 takes its maximum value is x = 1.

Verification of the Answer

Let's verify this result by substituting x = 1 into the function and comparing it with the values of the function for neighboring values of x.

When x = 0, f(0) = -(0)² + 2(0) - 1 = -1.

When x = 1, f(1) = -(1)² + 2(1) - 1 = 0.

When x = 2, f(2) = -(2)² + 2(2) - 1 = -1.

Therefore, we can see that when x = 1, the function f(x) = -x² + 2x - 1 takes its maximum value.

The value of x at which the function f(x) = -x² + 2x - 1 takes its maximum value is x = 1.