Question

Y=√x-x^2-как решить это неравенство?

Answer 1

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Answer 2

To solve the inequality Y = √x - x^2, we need to find the values of x that satisfy this inequality. To do this, we need to follow these steps:

Step 1: Find the domain of the function. The expression inside the square root must be non-negative (≥ 0), otherwise, the square root would be undefined for real numbers. So, we set the expression inside the square root greater than or equal to zero:

x - x^2 ≥ 0

Step 2: Solve for x. To solve this quadratic inequality, we first factor the expression:

x(1 - x) ≥ 0

Now, we need to find the critical points by setting each factor equal to zero:

x = 0 and 1 - x = 0

Solving for x in the second equation:

1 - x = 0 x = 1

Step 3: Create a sign chart to determine the solution intervals. We can create a sign chart to determine the sign of each factor and the sign of the entire expression (x - x^2).

Sign Chart: |0___________1_| x | - | + | + | 1 - x | + | 0 | - | x(1 - x)| 0 | - | 0 |

Step 4: Identify the solution intervals. The solution intervals are the regions where the expression x(1 - x) ≥ 0 is positive (greater than or equal to zero).

From the sign chart, we can see that x(1 - x) is greater than or equal to zero in two intervals:

1. 0 ≤ x < 1
2. x > 1

Step 5: Write the solution to the inequality. Now, let's rewrite the original inequality Y = √x - x^2.

√x - x^2 ≥ 0

Since Y = √x - x^2, we can write:

Y ≥ 0

So, the solution to the inequality Y = √x - x^2 is Y ≥ 0, and the corresponding intervals for x are 0 ≤ x < 1 and x > 1.