Question

Find Where Increasing/Decreasing Using Derivatives f(x)=x³-75x+3

Answer 1

To determine where the function is increasing or decreasing, we need to find the derivative of the function. The critical points are x = -5 and x = 5. The function is decreasing to the left of x = -5 and increasing to the right of x = 5.

Step-by-step explanation:

The given function is f(x) = x³ - 75x + 3. To determine where the function is increasing or decreasing, we need to find the derivative of the function. Taking the derivative of f(x), we get f'(x) = 3x² - 75. Now, we set f'(x) = 0 and solve for x:

1. 3x² - 75 = 0
2. 3x² = 75
3. x² = 25
4. x = ±5

So, the critical points of the function are x = -5 and x = 5. To determine if the function is increasing or decreasing at these points, we can test values on either side of the critical points. For example, if we choose x = -6, we plug it into f'(x) and get f'(-6) = 3(-6)² - 75 = -63. Since f'(-6) is negative, the function is decreasing to the left of x = -5. If we choose x = 4, we plug it into f'(x) and get f'(4) = 3(4)² - 75 = 3. Since f'(4) is positive, the function is increasing to the right of x = 5.