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Increasing/Decreasing Functions

Increasing/Decreasing Functions. As x increases y decreases. As x increases y increases. This function is decreasing when x < 0. This function is increasing when x > 0. Remember: First derivative gives a formula for the

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Increasing/Decreasing Functions

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  1. Increasing/Decreasing Functions As x increases y decreases As x increases y increases This function is decreasing when x < 0 This function is increasing when x > 0

  2. Remember: First derivative gives a formula for the slope of the tangent line on a curve. slope is negative graph is decreasing slope is positive graph is increasing Test for Increasing or Decreasing Functions If f ′(x) > 0 for all x in an interval I, then f is increasing on I. If f ′(x) < 0 for all x in an interval I, then fis decreasingon I.

  3. EXAMPLE 1 Calculus Approach Find the interval on which the function f(x) = x2 + 2x – 3is increasing and decreasing. Step 1: Find the derivative f ′(x) = 2x + 2 Step 2: Solve the inequalities 2x + 2 < 0 2x + 2 > 0 x > –1 x < –1 The function is decreasing when x < –1 The function is increasing when x > –1

  4. EXAMPLE 1 Graphing Approach Find the interval on which the function f(x) = x2 + 2x – 3is increasing and decreasing. Critical Value f ' (x) = 2x + 2 2x + 2 = 0 x = –1 f(–1) = (–1)2+ 2(–1) – 3 f(–1)= = – 4 Turning point The function is increasing when x > –1 The function is decreasing when x < –1

  5. EXAMPLE 2 f(x) = 2x3 – 6x2 Critical values of xoccur where the tangent has a slope of 0. f ′(x) = 6x2 – 12x The function increases when 6x(x – 2) > 0 x < 0 or x > 2 6x2 – 12x = 0 6x(x – 2) = 0 The function decreases when 6x(x – 2) < 0 0 < x < 2 x = 0 or x = 2 + – + 0 2

  6. f(x) = 2x3– 6x2 turning points decreasing 0 < x < 2 increasing x > 2 increasing x < 0

  7. EXAMPLE 3 Find the interval on which the function f(x) = x3 + 2x2 – 5x + 5is increasing and decreasing. Step 1: Find the derivative f ′ (x) = x2 + 4x – 5 Step 2: Factor the derivative f ′ (x) = x2 + 4x – 5 f ′ (x) = (x + 5)(x – 1)

  8. EXAMPLE 3 Find the interval on which the function f(x) = x3 + 2x2 – 5x + 5is increasing and decreasing. Step 3: Solve the inequalities (x + 5 )(x – 1) < 0 (x + 5)(x – 1) > 0 + – + – 5 1 The function is increasing when x< – 5 or x > 1 The function is decreasing when –5 < x < 1

  9. f(x) = x3 + 2x2 – 5x + 5 increasing x > 1 increasing x < –5 decreasing –5 < x < 1

  10. Rational Functions Remember Limits and the asymptotes on the graph? DNE Vertical asymptote x = 6 • Horizontal asymptote y = –1

  11. Example 4 Rational Functions Find the derivative This function always increasesbecause is always positive.

  12. x = 6 y = -1 increasing

  13. EXAMPLE 5 always positive 4x > 0 x > 0 4x < 0 x < 0 Increasing when Decreasing when

  14. EXAMPLE 5 decreasing increasing

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