# First Derivative Test

Definition:

Let $f(x)$ be a differentiable function and let $(a, f(a))$ be a critical point of $f(x)$.

1. If $f'(a) > 0$ to the left of $a$, and $f'(a) < 0$ to the right of $a$, then $(a, f(a))$ is a local minimum of $f(x)$.

2. If $f'(a) < 0$ to the left of $a$, and $f'(a) > 0$ to the right of $a$, then $(a, f(a))$ is a local maximum of $f(x)$.

3. In any other case, the critical point is neither a maximum or minimum. We call it a saddle point.

Example: Let $f(x) = 3x^5 - 5x^2$. Find all critical points of $f(x)$ and use the first derivative test to classify each point.

Solution:  First, we find the derivative: $f'(x) = 15x^4 - 15x^2$. We set the derivative equal to zero to find the critical points:

So, the critical points are (−1,2), (0,0), and (1,−2). We apply the first derivative test by finding the derivatives of points to the left and the right of each critical point:

So what does this tell us? Well we look at each point individually.

For $(-1, 2)$, the derivative changes from positive to negative. This tells us that $(-1, 2)$ is a local maximum.

For $(0,0)$, the derivative is negative both to the left and to the right of the point. This tells us that $(0,0)$ is a saddle point.

For $(1, -2)$, the derivative changes from negative to positive. This tells us that $(1, -2)$ is a local minimum.

We can examine the graph and confirm our findings:

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