Intervals of Increase and Decrease - Worksheets - 04 - 04

Q: What steps are required to solve each interval of increase/decrease problem?

Identify the x-coordinates of all critical points, find all discontinuities, then determine the open intervals where the function is increasing and where it is decreasing.

Q: How does the worksheet determine whether f + g is increasing on an interval?

It uses the fact that if x1 < x2 and both f and g are increasing, then f(x1) < f(x2) and g(x1) < g(x2), which implies f(x1)+g(x1) < f(x2)+g(x2).

Q: What example is given for functions that are increasing but whose difference is decreasing?

One example stated is f = x and g = 2x, where both functions are increasing on (−∞, ∞) but f − g decreases.

Oct 8, 2026

Upload by Quinn Holloway

Intervals of Increase and Decrease worksheets from Infinite Calculus focusing on analyzing functions using critical points and discontinuities. For each given function, the worksheet requires determining x-coordinates of critical points, listing discontinuities, and then identifying open intervals where the function is increasing and decreasing. It includes multiple polynomial examples, a rational expression with a discontinuity, and a section on critical thinking that proves how sums preserve monotonicity and asks for a counterexample involving differences.

LanguageEnglish

CategoryExam

FormatPDF

Pages4

Reading Time10 min

AccessFree to download

Intervals of Increase and Decrease

Problems 1-7

Critical Thinking Questions 8-9

Kuta Software-Infinite Calculus Name Intervals of Increase and Decrease Date Period For each problem, find the x-coordinates of all critical points, find all discontinuities, and find the open intervals where the function is increasing and decreasing. 1) y = −x3 + 2x2 + 2 2) y = x 3 − 11x2 + 39x − 47 3) y = −x4 + 3x2 − 3 4) y = 4xx~~ ~~4 3x2 − 3 5) y = ~~ ~~ x 3 2 6) y = (2x − 8) 3 5 2 1 3 3 7) y = − 5 (x − 4) − 2(x − 4) − 1 Critical thinking question: 8) If functions f and g are increasing on an interval, show that f + g is increasing on the same interval. 9) Give an example where functions f and g are increasing on the interval (−∞, ∞), but where f − g is decreasing. Kuta Software-Infinite Calculus Name Intervals of Increase and Decrease Date Period For each problem, find the x-coordinates of all critical points, find all discontinuities, and find the open intervals where the function is increasing and decreasing. 1) y = −x3 + 2x2 + 2 Critical points at: x = 0, 43 No discontinuities exist. Increasing: (0, 43) Decreasing: (−∞, 0),(~~ ~~43~~ ~~ , ∞) 2) y = x 3 − 11x2 + 39x − 47 Critical points at: x = 3, ~~ ~~133 No discontinuities exist. Increasing: (−∞, 3),(~~ ~~133~~ ~~ , ∞) Decreasing: (3, ~~ ~~133) 3) y = −x4 + 3x2 − 3 Critical points at: x = − ~~ ~~2~~ ~~, 0, ~~ ~~2~~ ~~ No discontinuities exist. Increasing: (−∞, −~~ ~~26~~ ~~ ),(0,~~ ~~26 ) Decreasing: (−~~ ~~26 , 0),(~~ ~~26 , ∞) 4) y = 4xx~~ ~~4 Critical points at: x = −2, 0 Discontinuity at: x = −1 Increasing: (−∞, −2), (0, ∞) Decreasing: (−2, −1), (−1, 0) 3x2 − 3 5) y = ~~ ~~ x 3 Critical points at: x = − 3 , 3 Discontinuity at: x = 0 Increasing: (− 3 , 0), (0, 3 ) Decreasing: (−∞, − 3 ), ( 3 , ∞) 2 6) y = (2x − 8) 3 Critical point at: x = 4 No discontinuities exist. Increasing: (4, ∞) Decreasing: (−∞, 4) 5 2 1 3 3 7) y = − 5 (x − 4) − 2(x − 4) − 1 Critical points at: x = 0, 4 No discontinuities exist. Increasing: (0, 4) Decreasing: (−∞, 0), (4, ∞) Critical thinking question: 8) If functions f and g are increasing on an interval, show that f + g is increasing on the same interval. We know that if x1 < x2 , then f (x1) < f (x2) and g(x1) < g(x2) . Therefore, f (x1) + g(x1) < f (x2) + g(x2) . 9) Give an example where functions f and g are increasing on the interval (−∞, ∞), but where f − g is decreasing. Many answers. Ex: f = x and g = 2x Create your own worksheets like this one with Infinite Calculus. Free trial available [at KutaSoftware.com](at KutaSoftware.com)

FAQs

What steps are required to solve each interval of increase/decrease problem?

Identify the x-coordinates of all critical points, find all discontinuities, then determine the open intervals where the function is increasing and where it is decreasing.

How does the worksheet determine whether f + g is increasing on an interval?

It uses the fact that if x1 < x2 and both f and g are increasing, then f(x1) < f(x2) and g(x1) < g(x2), which implies f(x1)+g(x1) < f(x2)+g(x2).

What example is given for functions that are increasing but whose difference is decreasing?

One example stated is f = x and g = 2x, where both functions are increasing on (−∞, ∞) but f − g decreases.

Oct 8, 2026

Upload by Quinn Holloway