Second Derivative Test Math

The test has three outcomes. If f x is positive at a critical point the function has a relative maximum at that point. So you fall back onto your first derivative.

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If f x is 0 at a critical point then the test is inconclusive.

Second derivative test math. If f x is negative at a critical point the function has a relative minimum at that point. At that point the second derivative is 0 meaning that the test is inconclusive. If then has a local minimum at. If the second derivative is greater than zero the stationary point is a minimum.

If the 2nd derivative is less than zero then the graph of the function is concave down. If the second derivative is less than zero the stationary point is a maximum. Now the second derivate test only applies if the derivative is 0. This means the second derivative test applies only for x 0.

If the 2nd derivative is greater than zero then the graph of the function is concave up. Suppose is a function of that is twice differentiable at a stationary point. The second derivative test for concavity states that. The extremum test gives slightly more general conditions under which a function with is a maximum or minimum.

X x values from the stationary points into the second derivative to perform the test. Inflection points indicate a change in concavity. If then has a local maximum at. The second derivative is often denoted as f x where the first derivative is denoted as f x.