https://file.aiquickdraw.com/mathbot/file/1726763505719enu4iagb.jpg

знайдіть проміжки зростання та спадання фунції

To find the intervals of increase and decrease for the function given:

\[
y = x^2 - 4x
\]

### Step 1: Find the first derivative
We take the derivative of the function \( y = x^2 - 4x \) to find the slope of the function at any point.

\[
\frac{dy}{dx} = 2x - 4
\]

### Step 2: Set the first derivative equal to zero
Setting the derivative equal to zero will help us find critical points, which are potential points where the function changes from increasing to decreasing or vice versa.

\[
2x - 4 = 0
\]

Solving for \( x \):

\[
2x = 4
\]
\[
x = 2
\]

### Step 3: Analyze the intervals
Now, we will analyze the behavior of the derivative on either side of \( x = 2 \). We choose test points from the intervals \( (-\infty, 2) \) and \( (2, \infty) \).

- For \( x < 2 \) (e.g., \( x = 0 \)):
  \[
  2x - 4 = 2(0) - 4 = -4
  \]
  The derivative is negative, so the function is decreasing in this interval.
  
- For

Learn how to find the intervals of increase and decrease for the quadratic function y = x^2 - 4x using the first derivative. This problem introduces the first derivative test, finding critical points, and analyzing the function's behavior. Suitable for Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation