Wize AP Calculus (BC) Textbook > Applications of Differentiation: Analytical

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

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

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

A function f(x)f(x)f(x)is decreasing if f′(x)<0\boxed{f'(x)<0}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^2f(x)=x2 is increasing or decreasing.

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

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


Therefore, f(x)f(x)f(x) is decreasing on the interval (−∞,0)(-\infin,0)(−∞,0) and increasing on the interval (0,∞)(0,\infin)(0,∞)

Find the intervals in which the function f(x)=−1x+1\displaystyle f(x)=\frac{-1}{x+1}f(x)=x+1−1​ is increasing.

(−∞,∞)\left(-\infty,\infty\right)(−∞,∞)

(−∞,1)∪(1,∞)\left(-\infty,1\right)\cup\left(1,\infty\right)(−∞,1)∪(1,∞)

(−∞,−1)\left(-\infty,-1\right)(−∞,−1)

(−∞,−1)∪(−1,∞)\left(-\infty,-1\right)\cup\left(-1,\infty\right)(−∞,−1)∪(−1,∞)

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