The general equation for determining where a function is increasing or decreasing is the first derivative test, where you take the derivative of the function and set it equal to zero to find critical values. These critical values divide the function into different intervals, where the function is either increasing or decreasing within each interval.
To use the first derivative test, you take the derivative of the function and set it equal to zero. Then, you solve for x to find the critical values. Next, you create a number line with the critical values and test a value within each interval to see if the derivative is positive or negative. If the derivative is positive, the function is increasing in that interval. If the derivative is negative, the function is decreasing in that interval.
Yes, x^3 + 12x + 5 can have both increasing and decreasing intervals. This is because the function can have multiple critical values, which creates different intervals where the function can either be increasing or decreasing.
To determine if a critical value is a local maximum or minimum, you can use the second derivative test. You take the second derivative of the function and plug in the critical value. If the second derivative is positive, the critical value is a local minimum. If the second derivative is negative, the critical value is a local maximum.
The graph of x^3 + 12x + 5 reflects its increasing and decreasing intervals by having a positive slope in the increasing intervals and a negative slope in the decreasing intervals. The critical values also represent points where the graph changes direction from increasing to decreasing or vice versa.