Increasing and Decreasing Functions

Mathematics · Calculus & Analysis · ages 17-18

Name ______________________ Date ____________

Is the line y = 5x + 2 increasing, decreasing, or neither?
- decreasing for all x
- neither, because it is a straight line
- increasing for all x
- increasing only for positive x
If f'(x) is negative at every point of an interval, the function is decreasing across that whole interval.
Circle one: True False
For which values of x is y = 3 - 2x decreasing?
- for x > 0 only
- for all values of x
- for x < 1.5 only
- for no values of x at all
On which interval is y = x² - 6x decreasing?
- x < 3
- x > 3
- x < 6
- all values of x
The curve y = 2x² - 16x + 5 stops decreasing at one value of x. What is it?
Answer: ______________
The curve y = x³ + 4x is increasing for every value of x.
Circle one: True False
For which values of x is y = x³ + 3x² increasing?
- x < -2 or x > 0
- -2 < x < 0
- only x < 0 or x > 2
- for all x
Noah claims y = x³ + x decreases somewhere. Is Noah right?
Circle one: True False
For which values of x is y = 2x³ + 3x² - 12x increasing?
- x < -2 or x > 1
- -2 < x < 1
- x < -1 or x > 2
- for all x
Mia differentiates y = x³ - 48x, solves 3x² - 48 > 0 to get x² > 16, and writes the answer as x > 4. Where is the slip?
- the inequality sign flips when you take the square root, so the answer is x < 4
- the derivative should have been 3x² + 48
- she should have solved 3x² - 48 < 0 instead
- x² > 16 also holds for x < -4, so the answer has two parts

Answer key

For grown-ups. Fold this page away before handing over the rest.

increasing for all x · dy/dx = 5, which is positive for every x, so the line rises everywhere. A straight line is a perfectly ordinary function here.
True · A negative derivative at a point means the curve is falling there. Holding at every point of the interval means it falls all the way across.
for all values of x · dy/dx = -2, which is negative for every x, so this line falls everywhere.
x < 3 · f'(x) = 2x - 6, which is negative exactly when x < 3.
4 · dy/dx = 4x - 16, which is zero at x = 4. Below 4 it is negative and above 4 it is positive.
True · dy/dx = 3x² + 4. A square is never negative, so 3x² is at least 0 and the whole derivative is at least 4, which is positive everywhere.
x < -2 or x > 0 · dy/dx = 3x² + 6x = 3x(x + 2), which is zero at 0 and -2. An upward parabola is positive outside its roots, so the curve rises for x < -2 and for x > 0.
False · Noah is wrong: f'(x) = 3x² + 1 is always positive, so y = x³ + x increases everywhere.
x < -2 or x > 1 · dy/dx = 6x² + 6x - 12 = 6(x + 2)(x - 1), which is zero at -2 and 1. An upward parabola is positive outside its roots.
x² > 16 also holds for x < -4, so the answer has two parts · A square bigger than 16 means the number is further than 4 from zero in either direction. Testing x = -5 gives 25 > 16, so that stretch counts too.