Extrema and Increasing Decreasing Intervals

Front

What is an increasing interval in a function?

Back

An increasing interval is a range of x-values where the function's output (y-values) is rising as x increases.

Front

What is a decreasing interval in a function?

Back

A decreasing interval is a range of x-values where the function's output (y-values) is falling as x increases.

Front

How do you determine increasing and decreasing intervals from a graph?

Back

By analyzing the slope of the tangent line: if the slope is positive, the function is increasing; if negative, it is decreasing.

Front

What is an extremum in calculus?

Back

An extremum is a point in a function where it reaches a local maximum or minimum value.

Front

What is the difference between local maximum and local minimum?

Back

A local maximum is the highest point in a neighborhood, while a local minimum is the lowest point in a neighborhood.

Front

How can you find extrema using the first derivative test?

Back

By finding critical points where the first derivative is zero or undefined, and then testing intervals around these points.

Front

What does it mean if a function has no extrema?

Back

It means the function does not have any local maximum or minimum points within the given interval.

Front

What is the significance of the second derivative in determining concavity?

Back

The second derivative indicates concavity: if positive, the function is concave up; if negative, concave down.

Front

How do you find the intervals of increase or decrease using the first derivative?

Back

By analyzing the sign of the first derivative: positive indicates increasing, negative indicates decreasing.

Front

What is the role of critical points in finding extrema?

Back

Critical points are potential locations for extrema, where the first derivative is zero or undefined.