Question

задана функция у = - х квадрат + 8 х + с определите значение с при котором наибольшее значение функции с равно 7

Answer 1

C=-9
находим вершину параболы -b=-8 a=-1
x0=8/(-2)=4
находим значение функции
y(4)=-16+32+c=7
16+c=7
c=7-16=-9

Answer 2

Finding the Maximum Value of a Function

To find the value of c for which the function u = -x^2 + 8x + c has its maximum value equal to 7, we can use the concept of completing the square to find the vertex of the parabola represented by the function.

The vertex form of a parabola is given by y = a(x - h)^2 + k, where (h, k) is the vertex of the parabola.

By completing the square, we can rewrite the function u in vertex form and then determine the value of c.

Completing the Square

The function u = -x^2 + 8x + c can be rewritten in the form u = -(x^2 - 8x) + c.

To complete the square, we need to add and subtract (8/2)^2 = 16 inside the parentheses, which gives us u = -(x^2 - 8x + 16 - 16) + c.

Simplifying, we get u = -(x^2 - 8x + 16) + 16 + c.

Now, we can rewrite the expression inside the parentheses as a perfect square: u = -(x - 4)^2 + 16 + c.

Finding the Maximum Value

From the vertex form, we can see that the maximum value of u occurs when (x - 4)^2 = 0, which means x = 4.

Substituting x = 4 into the function, we get u = -(4 - 4)^2 + 16 + c = 16 + c.

Given that the maximum value of u is 7, we can solve for c:

16 + c = 7 c = 7 - 16 c = -9

Therefore, the value of c for which the maximum value of the function u is equal to 7 is c = -9.