# 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: