How do you use the second derivative test to find the relative maxima and minima of the given #f(x)= x^4 - (2x^2) + 3#?
A relative maximum is where the first derivative is null and the second derivative is negative.
A relative minimum is where the first derivative is null and the second derivative is positive.
We want the curve's relative maxima and minima, that is, where the curve stops increasing to start decreasing or vice-versa. At these points, the curve's slope will be null.
Let's look for the solutions to
We have three solutions here:
Now we want to know if these points are maxima or minima.
If a point is a maximum , the curve will be increasing before reaching the point and be decreasing after passing the point.
If a point is a minimum , the curve will be decreasing before reaching the point and be increasing after passing the point.
When a curve is increasing, its slope is positive.
When a curve is decreasing, its slope is negative.
So we want to know if, at a given point, the slope (first derivative) is:
To do so, we use the second derivative: