Features Of Polynomials

1. Statement: An absolute (global) maximum is also a relative (local) maximum.

True

False

2. Statement: At a location where the first derivative of a function is zero, the function may have a local minimum, a local maximum, or a point of inflection.

True

False

3. Match the statements about derivatives to the corresponding features.

f'(x) > 0

f'(x) < 0

f''(x) > 0

f''(x) < 0

Submit

4. For a polynomial P(x) defined on an interval [a,b], where do you need to check for minima and maxima?

Wherever P(x) = 0

Wherever P'(x) = 0

Wherever P''(x) = 0

At x = a and x = b

Submit

5. Statement: Relative (local) extremes cannot occur at endpoints unless the slope of the tangent line at the endpoint is zero.

True

False

6. Consider the polynomial f(x) and the sign line shown for its derivative f'(x). Which statement about increasing vs. decreasing is true?

The function f is increasing on the interval [–1,1]

The function f is decreasing on the inverval [–1,1]

The function f is decreasing on the interval [0,5/2]

The function f is overall an increasing function

7. Statement: The slope of the tangent line at a point of inflection is always zero.

True

False

8. Statement: A point of inflection is found at any location where the second derivative of a function is zero.

True

False

9. Consider the polynomial f(x) and the sign line shown for its derivative f'(x). Which statement best describes the relative extrema of f?

The function f has relative minima at –2 and 5/2, and a relative maximum at 0

The function f has a relative minimum at 0 and a relative maximum at 5/2

The function f only has one relative extreme: a maximum at –2

The function f has a relative maximum at 0 and a relative minimm at 5/2

10. Consider the polynomial f(x) and the sign line shown for its derivative f'(x). Which statement about inflection points of f is definitely true?

The function f does not have any inflection points

The function f has an inflection point at –2 and possible inflection points at 0 and/or 5/2

The function f has an inflection point at –2 and potentially other inflection points, but NOT at 0 or 5/2

The function f has exactly four inflection points