Question

Як знаходити найбільше значення х через квадрат суми та квадрат різниці 5х-х²-6 без вершини параболи, графіка функції, диференціювання функції та дескримінантів (7 клас) ​

Answer 1

Ответ:

Объяснение:

Выделяем квадрат, а после можно наглядно увидеть как размещена парабола.

Answer 2

Finding the Maximum Value of a Quadratic Function

To find the maximum value of the quadratic function f(x) = 5x - x² - 6 without using the vertex of the parabola, we can use the method of completing the square. This involves rewriting the quadratic function in a form that allows us to easily identify the maximum value.

First, let's rewrite the function in the standard form by completing the square:

f(x) = -x² + 5x - 6

To find the maximum value, we can use the formula for the vertex of a parabola in the form f(x) = ax² + bx + c:

Vertex x-coordinate (h) = -b/2a

Vertex y-coordinate (k) = f(h)

In this case, the x-coordinate of the vertex is given by h = -b/2a and the y-coordinate is given by k = f(h).

Let's calculate the x-coordinate of the vertex:

h = -b/2a = -5 / (2 * (-1)) = 5 / 2

Now, we can find the y-coordinate of the vertex by substituting the x-coordinate into the function:

k = f(5/2) = -(5/2)² + 5 * (5/2) - 6

k = -25/4 + 25/2 - 6

k = -25/4 + 50/4 - 24/4

k = 1/4

So, the maximum value of the function f(x) = 5x - x² - 6 is 1/4.

Graph of the Function

The graph of the function f(x) = 5x - x² - 6 will be a parabola. By finding the vertex as we did above, we can determine that the vertex is at the point (5/2, 1/4). This means the parabola opens downwards and the vertex represents the maximum point on the graph.

Differentiation of the Function

We can also find the maximum value of the function by differentiating it and finding the critical points. The critical points occur where the derivative is equal to zero or undefined. By setting the derivative of the function equal to zero and solving for x, we can find the x-coordinate of the maximum point.

The derivative of the function f(x) = 5x - x² - 6 is:

f'(x) = 5 - 2x

Setting the derivative equal to zero:

5 - 2x = 0

Solving for x:

2x = 5

x = 5/2

This is the same x-coordinate of the vertex that we found earlier.

Discriminant

The discriminant of a quadratic function can provide information about the nature of its roots. For the quadratic function f(x) = 5x - x² - 6, the discriminant is given by:

Δ = b² - 4ac

Where a = -1, b = 5, and c = -6.

Substituting these values:

Δ = 5² - 4*(-1)*(-6)

Δ = 25 - 24

Δ = 1

Since the discriminant is positive, the quadratic function has two distinct real roots, and the maximum value occurs at the vertex of the parabola.

I hope this helps! If you have any further questions or need additional assistance, feel free to ask.