2nd Derivative Applications & Derivative Graphs - Stinnett copy

Front

What is the second derivative test for local extrema?

Back

The second derivative test states that if f''(x) > 0 at a critical point, then f has a local minimum at that point. If f''(x) < 0, then f has a local maximum. If f''(x) = 0, the test is inconclusive.

Front

Define relative maximum and minimum in the context of a function's graph.

Back

A relative maximum is a point where the function's value is higher than all nearby points, while a relative minimum is a point where the function's value is lower than all nearby points.

Front

What does it mean if the first derivative of a function is zero?

Back

If f'(x) = 0, it indicates a critical point, which could be a local maximum, local minimum, or a point of inflection.

Front

Explain the significance of the first derivative test.

Back

The first derivative test helps determine whether a critical point is a local maximum or minimum by analyzing the sign of f' before and after the critical point.

Front

What is concavity in relation to a function's graph?

Back

Concavity refers to the direction the graph of a function curves. A function is concave up if its graph opens upwards and concave down if it opens downwards.

Front

How do you determine intervals of concavity using the second derivative?

Back

To determine intervals of concavity, find the second derivative f''(x). If f''(x) > 0, the function is concave up on that interval; if f''(x) < 0, it is concave down.

Front

What is a point of inflection?

Back

A point of inflection is where the concavity of a function changes, which occurs when f''(x) = 0 or is undefined.

Front

State the relationship between the first and second derivatives at a local maximum.

Back

At a local maximum, the first derivative f'(x) = 0 and the second derivative f''(x) < 0.

Front

What does it mean if the second derivative is positive?

Back

If f''(x) > 0, the function is concave up at that point, indicating that the slope of the tangent line is increasing.

Front

What does it mean if the second derivative is negative?

Back

If f''(x) < 0, the function is concave down at that point, indicating that the slope of the tangent line is decreasing.