Nova AP Calc Maxima and Minima

Vertical Asymptotes

A line x = k is a vertical asymptote of the graph of y = f(x) if the limit of f(x) as x approaches k⁺ equals ±∞ or if the limit of f(x) as x approaches k⁻ equals ±∞.

Horizontal Asymptotes

A line y = c is a horizontal asymptote of the graph of y = f(x) if the limit of f(x) as x approaches ∞ equals c or if the limit of f(x) as x approaches -∞ equals c.

What is a relative maximum or minimum?

A relative or local maximum or minimum means that the curve has a horizontal tangent line at that point, but it is not the highest or lowest value that the function attains.

What is an absolute maximum or minimum?

An absolute maximum or minimum means that the curve has attained its highest or lowest value. It occurs either at an artificial point or an end point.

Second Derivative Test

If a function has a critical value at x = c, then that value is a relative maximum if f''(c) < 0 and it is a relative minimum if f''(c) > 0.

Finding a Cusp

If the derivative of a function approaches ∞ from one side of a point and -∞ from the other, and if the function is continuous at that point, then the curve has a cusp at that point.

Concavity Test/First Derivative Test

When f''(x) > 0, the curve is concave up; when f''(x) < 0, the curve is concave down; when f''(x) = 0, the curve is at a point of inflection.

End Behavior of a Function

When f'(x) > 0, the curve is rising; when f'(x) < 0, the curve is falling; when f'(x) = 0, the curve is at a critical point.

Where does a maximum or minimum of a function occur?

A maximum or minimum of a function occurs at a point where the derivative of a function is zero, or where the derivative fails to exist.