Relative Maximum and Minimum of a Polynomial

Front

What is a relative maximum in a polynomial function?

Back

A relative maximum is a point on the graph of a polynomial function where the function value is greater than the values of the function at nearby points.

Front

What is a relative minimum in a polynomial function?

Back

A relative minimum is a point on the graph of a polynomial function where the function value is less than the values of the function at nearby points.

Front

How do you find relative maximums and minimums in a polynomial function?

Back

Relative maximums and minimums can be found at the turning points of the function, which are points where the derivative changes sign.

Front

What is a turning point in the context of polynomial functions?

Back

A turning point is a point on the graph where the function changes from increasing to decreasing (maximum) or from decreasing to increasing (minimum).

Front

What is the significance of the first derivative in finding relative extrema?

Back

The first derivative of a function indicates the slope of the function. Setting the first derivative to zero helps identify potential relative maximums and minimums.

Front

What does the second derivative test determine?

Back

The second derivative test helps determine the concavity of the function at a critical point. If the second derivative is positive, the point is a relative minimum; if negative, it is a relative maximum.

Front

What is the relationship between the degree of a polynomial and the number of relative extrema?

Back

The maximum number of relative extrema of a polynomial function is at most one less than its degree.

Front

What is an x-intercept of a polynomial function?

Back

An x-intercept is a point where the graph of the polynomial crosses the x-axis, meaning the function value is zero at that point.

Front

How can you determine the number of relative maximums and minimums from a polynomial's graph?

Back

By analyzing the turning points on the graph, you can count the number of relative maximums and minimums.

Front

What does it mean if a polynomial function has no relative maximums?

Back

If a polynomial function has no relative maximums, it means that the function either increases or decreases without bound in its domain.