find the increasing and decreasing intervals of 4x^3-7x+3
To find the increasing and decreasing intervals of the function f(x) = 4x^3 - 7x + 3, we need to determine where the function is increasing or decreasing.
Step 1: Find the derivative of the function f(x) with respect to x.
The derivative of f(x) will give us information about the slope of the function at each point.
f '(x) = 12x^2 - 7
Step 2: Set f '(x) = 0 and solve for x to find the critical points.
Critical points occur where the derivative is equal to zero or undefined.
12x^2 - 7 = 0
12x^2 = 7
x^2 = 7/12
x = ± √(7/12)
Step 3: Create a number line and identify the intervals between the critical points.
By plugging in test values into f '(x) in each interval, we can determine whether the function is increasing or decreasing in that interval.
Number line: -∞, -√(7/12), √(7/12), +∞
Consider a value slightly less than -√(7/12), such as -1. Plug this value into f '(x):
f '(-1) = 12(-1)^2 - 7 = 5
Since f '(-1) > 0, this means f(x) is increasing to the left of -√(7/12).
Now consider a value between -√(7/12) and √(7/12), such as 0. Plug this value into f '(x):
f '(0) = 12(0)^2 - 7 = -7
Since f '(0) < 0, this means f(x) is decreasing in this interval.
Finally, consider a value slightly greater than √(7/12), such as 1. Plug this value into f '(x):
f '(1) = 12(1)^2 - 7 = 5
Since f '(1) > 0, this means f(x) is increasing to the right of √(7/12).
Therefore, the increasing interval is (-∞, -√(7/12)) ∪ (√(7/12), +∞), and the decreasing interval is (-√(7/12), √(7/12)).