Question

Функция y=-x^2+bx+c при x=-1 принимает наибольшее значение, равное 5. Найдите y(1)

Answer 1

x=-1=х₀ --это абсцисса вершины параболы

х₀ = -b\(2a)

-1 = -b\(2*(-1)) = -b\(-2) ---> -b = 2 ---> b = -2

при х=-1: у(-1) = -(-1)²+b*(-1)+c = 5

5 = -1-b+c ---> 5 = -1+2+c ---> c = 4

y(1) = -1²+b+c = -1-2+4 = 1

Answer 2

To find the value of y(1), we need to first determine the values of b and c based on the given information.

Given that the function y = -x^2 + bx + c takes its maximum value at x = -1 and the maximum value is 5, we can use this information to find the values of b and c.

When the function has a maximum value at x = -1, its vertex is located at that point. The vertex form of a quadratic function is given by:

y = a(x - h)^2 + k

where (h, k) is the vertex of the parabola.

In this case, h = -1 because the maximum occurs at x = -1.

So, the equation becomes:

y = a(x + 1)^2 + k

We know that y = 5 when x = -1:

5 = a(-1 + 1)^2 + k 5 = a(0) + k 5 = k

Now, we have k = 5.

Next, let's find the value of b:

We know that the coefficient of x in the vertex form of the quadratic is b. Therefore, b = -2a (by comparing it with y = ax^2 + bx + c).

Since the function passes through the point (1, y), we can use this information to find the value of a:

y = -x^2 + bx + c y = -(1)^2 + b(1) + c y = -1 + b + c

We also know that y = 5 when x = -1:

5 = -1 + b + c

Now, we can substitute k = 5 into the equation:

5 = -1 + b + 5 5 = b + 4 b = 5 - 4 b = 1

Now, we have the values of b and k:

b = 1 k = 5

So, the quadratic function becomes:

y = -x^2 + bx + c y = -x^2 + 1x + 5

Now, let's find y(1) by substituting x = 1 into the function:

y = -(1)^2 + 1(1) + 5 y = -1 + 1 + 5 y = 5

Therefore, y(1) is equal to 5.