Final answer:
To determine where a function is increasing and decreasing, analyse the function's slope and its first derivative. If the slope or first derivative is positive, the function is increasing. If negative, the function is decreasing.
Step-by-step explanation:
To determine where a function is increasing and decreasing, we need to analyse its slope. If the slope of a function is positive, the function is increasing. If the slope is negative, the function is decreasing. To find the increasing and decreasing intervals, we can also look at where the first derivative of the function is positive or negative. If the first derivative is positive, the function is increasing, and if the first derivative is negative, the function is decreasing.
Let's take an example:
If we have a function f(x) = x^2, we can find its derivative f'(x) = 2x. To determine the intervals where the function is increasing or decreasing, we can analyze the sign of the derivative. Since f'(x) = 2x is positive for x > 0, the function f(x) = x^2 is increasing for x > 0. And since f'(x) = 2x is negative for x < 0, the function f(x) = x^2 is decreasing for x < 0.