Let's say a function is given by f(x) = 3x² - 6x² +2x - 1. Find the first and second derivative of f(x). Using the first derivative determine whether f(x) is increasing or decreasing when x = 0 and x= 1
To find the first derivative of f(x), we differentiate the function f(x) = 3x² - 6x² + 2x - 1 with respect to x:
f'(x) = d/dx (3x²) - d/dx (6x²) + d/dx (2x) - d/dx (1)
f'(x) = 6x - 12x + 2
Simplify to get:
f'(x) = -6x + 2
To find the second derivative of f(x), we differentiate the first derivative with respect to x:
f''(x) = d/dx (-6x + 2)
f''(x) = -6
Now, we can use the first derivative to determine whether f(x) is increasing or decreasing at x = 0 and x = 1:
At x = 0:
f'(0) = -6(0) + 2 = 2
Since f'(0) is positive, f(x) is increasing at x = 0.
At x = 1:
f'(1) = -6(1) + 2 = -4
Since f'(1) is negative, f(x) is decreasing at x = 1.