Increasing and Decreasing

Learn derivative signs and monotonicity, negative derivative values.

Lesson 1 of 32 parts Practice Flashcards

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Derivative Signs and Monotonicity

Objective: Explain derivative signs and monotonicity, apply the stated method, and check its conditions.

On an interval where f is differentiable, f′>0 implies increasing and f′<0 implies decreasing. A critical input is a domain point where f′ is zero or undefined. Split intervals at candidates and test signs rather than inferring behavior from one zero alone. A zero derivative alone does not prove a turning point.

Formulas

$f′>0: increasing; f′<0: decreasing$

Worked examples

Try itFor f(x)=(x−7)², where does the sign of f′ change? Give the x boundary.Show answer

Step 1: f′(x)=2(x−7). Step 2: It is negative for x<7 and positive for x>7. Step 3: The change occurs at x=7.

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Derivative Signs and Monotonicity

Objective: Explain derivative signs and monotonicity, apply the stated method, and check its conditions.

Formulas

$f′>0: increasing; f′<0: decreasing$

Worked examples

Try itFor f(x)=(x−7)², where does the sign of f′ change? Give the x boundary.Show answer

Step 1: f′(x)=2(x−7). Step 2: It is negative for x<7 and positive for x>7. Step 3: The change occurs at x=7.

Negative Derivative Values

Objective: Explain negative derivative values, apply the stated method, and check its conditions.

A negative derivative indicates that outputs decrease locally as inputs increase. It does not mean the function value itself is negative. Distinguish slope sign from output sign, and use interval sign tests to establish decreasing behavior over a whole interval. A negative slope and a negative function value are different statements.

Formulas

f′(x)<0 indicates local decrease

Worked examples

Try itFor f(x)=−(x−7)², find f′(13).Show answer

Step 1: f′(x)=−2(x−7). Step 2: At x=13, the difference is 6. Step 3: f′=-12, so the function is decreasing there.

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