First Derivative Test

Front

What is the First Derivative Test used for?

Back

The First Derivative Test is used to determine the local maxima and minima of a function by analyzing the sign of its first derivative.

Front

What does it mean if f'(x) > 0 on an interval?

Back

If f'(x) > 0 on an interval, it means that the function f(x) is increasing on that interval.

Front

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

Back

If f'(x) < 0 on an interval, it means that the function f(x) is decreasing on that interval.

Front

What is the significance of f'(c) = 0 at a point c?

Back

If f'(c) = 0, it indicates that c is a critical point, which may be a local maximum, local minimum, or a point of inflection.

Front

How do you determine if a critical point is a local maximum or minimum using the First Derivative Test?

Back

To determine if a critical point is a local maximum or minimum, check the sign of f'(x) before and after the critical point. If f' changes from positive to negative, it's a local maximum; if it changes from negative to positive, it's a local minimum.

Front

What is a local minimum?

Back

A local minimum is a point where the function value is lower than the values of the function at nearby points.

Front

What is a local maximum?

Back

A local maximum is a point where the function value is higher than the values of the function at nearby points.

Front

What does it indicate if f'(x) is zero at a point but does not change sign?

Back

If f'(x) is zero at a point and does not change sign, that point is neither a local maximum nor a local minimum; it may be a point of inflection.

Front

What is the relationship between the first derivative and the slope of the tangent line?

Back

The first derivative f'(x) represents the slope of the tangent line to the graph of the function f(x) at the point x.

Front

What is the interval of increase for a function?

Back

The interval of increase for a function is the set of x-values where the function's first derivative is positive.