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Wize University Calculus 1 Textbook > Applications of Differentiation

Intervals of Increase and Decrease

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Intervals of Increase and Decrease

If a function has a positively sloped tangent line at a certain point, the function must be going up. Thus, the derivative tells us where functions are increasing and decreasing.

Intervals of Increase

A function f(x)f(x) is increasing on an interval if when x≤yx\leq y, then f(x)≤f(y)f(x)\le f(y) , for every xx and yy in that interval.

A function f(x)f(x) is increasing if f′(x)>0\boxed{f'(x)>0} on that interval.



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Intervals of Decrease

A function f(x)f(x) is decreasing on an interval if when x≤yx\leq y, then f(x)≥f(y)f(x)\ge f(y) , for every xx and yy in that interval.

A function f(x)f(x)is decreasing if f′(x)<0\boxed{f'(x)<0}on that interval.




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Example: Intervals of Increase and Decrease

Find the intervals in which the function f(x)=x2f\left(x\right)=x^2 is increasing or decreasing.

The derivative is f′(x)=2xf'\left(x\right)=2x

Find where f′(x)=0f'\left(x\right)=0 or is undefined:
2x=02x=0
⇒x=0\Rightarrow x=0


Therefore, f(x)f(x) is decreasing on the interval (−∞,0)(-\infin,0) and increasing on the interval (0,∞)(0,\infin)
Find the intervals in which the function f(x)=−1x+1\displaystyle f(x)=\frac{-1}{x+1} is increasing.
Extra Practice
Log & Exponential Function Properties
Which of the following functions is/are always increasing?
Intervals of Increase and Decrease
Find the intervals on which the function f(x)=−1x+1\displaystyle f(x)=-\frac{1}{x+1} is increasing or decreasing.
Intervals of Increase and Decrease
Find the intervals in which the function f(x)=x3+3x2−9x+1f(x)=x^{3}+3x^{2}-9x+1 is increasing or decreasing.
Q.\textbf{Q.} Find the absolute extrema of f(x)=2xx2+1\displaystyle f(x)=\frac{2x}{x^2+1} on the interval [0,2][0,2]. Then find the local/absolute extrema on its domain, and intervals of increase and decrease.
Intervals of Increase and Decrease
Find the intervals on which the function f(x)=x3+3x2−9x+1f(x)=x^{3}+3x^{2}-9x+1 is increasing or decreasing.
Q:\bf{Q:} Which of the following functions is/are always increasing?
Practice: Derivatives as a Rate of Change
f′(x)f'(x) plot for a differentiable function is depicted as below. Find interval(s) for which f(x)f\left(x\right) is increasing.
Intervals of Increase and Decrease
If the function is given by f(x)=∫x2(t+1)2(t−2)t dtf(x)=\int_x^2\left(t+1\right)^2\left(t-2\right)t\ dt, then f(x)f\left(x\right) is decreasing on the interval
Critical Points and Extrema
The following is the graph of f′′(x)f''\left(x\right), which of the following statements (if any) is correct?
Intervals of Increase and Decrease: Rate of Change
f′(x)f'(x) plot for a differentiable function is depicted as below. Find interval(s) for which f(x)f\left(x\right) is increasing.
Intervals of Increase and Decrease
f′(x)f'(x) plot for a differentiable function is depicted as below. Find interval(s) for which f(x)f\left(x\right) is increasing.
Concavity and Inflection Points
Find the intervals of increase/decrease and the intervals of concavity for f(x)=ln⁡(x2−1)f(x)=\ln(x^2-1)
Concavity, Inflection Points, Extrema and Intervals
Given f(x)=2−2x+13x3f(x)=2-2x+\frac{1}{3}x^3, find
the intervals of increase and decrease
the local maximum and minimum values
Increasing Interval: Relative extrema
If the derivative of the function ff is f′(x)=ex(x−3)2(x+1)(2−x)f'\left(x\right)=e^x\left(x-3\right)^2\left(x+1\right)\left(2-x\right), then f(x)f\left(x\right) is increasing on the interval?
Intervals of Increase and Decrease
Find the intervals in which the function f(x)=x3+3x2−9x+1f(x)=x^{3}+3x^{2}-9x+1 is increasing or decreasing.
Q.\textbf{Q.} Find the absolute extrema of f(x)=2xx2+1\displaystyle f(x)=\frac{2x}{x^2+1} on the interval [0,2][0,2]. Then find the local/absolute extrema on its domain, and intervals of increase and decrease.
Intervals of Increase and Decrease
Find the intervals on which the function f(x)=x3+3x2−9x+1f(x)=x^{3}+3x^{2}-9x+1 is increasing or decreasing.
Intervals of Increase and Decrease
Find the intervals on which the function f(x)=−1x+1\displaystyle f(x)=-\frac{1}{x+1} is increasing or decreasing.
Q:\bf{Q:} Which of the following functions is/are always increasing?
Optimization
For the following function, determine the intervals in which the function is increasing or decreasing, its critical points, and the intervals in which the function is concave upwards or downwards.
f(x)=x2−2ln⁡xf(x)=x^2-2\ln{x}
f′(x)f'(x) plot for a differentiable function is depicted below.
Find interval(s) at which ff is increasing.
f′(x)f'(x) plot for a differentiable function is depicted below.
a) Find all critical point of ff in the range shown in the plot.
b) Determine which of the critical points found in the previous part are local max/min.
Intervals of increase and decrease
For the following function, determine the intervals in which the function is increasing or decreasing, its critical points, and the intervals in which the function is concave upwards or downwards.
f(x)=x3ln⁡xf(x) = x^3 \ln x
Optimization
For the following function, determine the intervals in which the function is increasing or decreasing, its critical points, and the intervals in which the function is concave upwards or downwards.
y=e4x2xy=\frac{e^{4x}}{2x}
Log & Exponential Function Properties
Which of the following functions is/are always increasing?
Intervals of increase and decrease
If f′(x)=x5x+1(x−1)3(x+2)f'\left(x\right)=x^5\sqrt{x+1}\left(x-1\right)^3\left(x+2\right), which of the following statements about the graph of f(x)f\left(x\right) must be true?
(Select all that apply)
Exercises
Find the intervals in which the function f(x)=−1x+1\displaystyle f(x)=\frac{-1}{x+1} is increasing or decreasing.
f′(x)f'(x) plot for a differentiable function is depicted as below. Answer the following questions.
Find interval(s) at which ffis increasing.