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#?
1 Answer
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.
Explanation:
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:
negative-null-positive
or
positive-null-negative
To do so, we use the second derivative: