First Derivative Test Review

Front

What is the First Derivative Test?

Back

The First Derivative Test is a method used to determine local extrema (maximum and minimum points) of a function by analyzing the sign of the first derivative.

Front

What does it mean if f'(x) changes from positive to negative?

Back

It indicates that there is a local maximum at that point.

Front

What does it mean if f'(x) changes from negative to positive?

Back

It indicates that there is a local minimum at that point.

Front

If f'(x) < 0, what can be said about the function f(x)?

Back

The function f(x) is decreasing.

Front

If f'(x) = 0 at a point (a,b), what can be concluded?

Back

The point (a,b) is a candidate for a local extremum.

Front

What is the second derivative test?

Back

The second derivative test is used to determine the concavity of a function at a critical point. If f''(x) > 0, the function is concave up (local minimum); if f''(x) < 0, the function is concave down (local maximum).

Front

Find the first derivative of f(x) = x^2 - 6x + 5.

Back

f'(x) = 2x - 6.

Front

What is the minimum value of f(x) = x^2 - 6x + 5?

Back

The minimum value is -4.

Front

What is the second derivative of f(x) = 2x - 5x^6?

Back

f''(x) = -150x^4.

Front

What does it mean if f''(x) < 0?

Back

The function is concave down.