Worksheet on identifying relative extrema and intervals of increase and decrease for various functions, including polynomial and rational functions.

Let’s solve each problem step by step, approximating relative extrema and intervals of increase/decrease based on the graphs or using a graphing calculator as instructed.

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## Problems 1 & 2: Approximate relative extrema from graphs

Problem 1:

Graph shows a piecewise-like curve with:
- A sharp peak (cusp) at approximately x = -3, y ≈ -1
- A smooth valley (minimum) around x = 2, y ≈ -4

✔ Relative Maximum: At (-3, -1) — because it's a local high point (even though it's a cusp, not differentiable, it’s still a relative max).
✔ Relative Minimum: At (2, -4) — lowest point in its neighborhood.

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Problem 2:

Graph has a vertical asymptote at x = -1 (dashed line).

Left of x = -1: Curve comes down from top left → reaches a low point near x = -2.5, y ≈ 2.5? Wait — let’s look again.

Actually, to the left of x = -1, the graph is decreasing from upper left → reaches a local minimum around x = -2, y ≈ 2.

To the right of x = -1, the graph starts from negative infinity, rises to a local maximum around x = 0, y ≈ 0, then decreases.

Wait — actually, looking carefully:

- Left side (x < -1): The curve is decreasing from top-left toward x = -1. So no local min/max on left? Actually, it looks like it curves upward slightly before dropping — but no clear extremum. Let’s re-express.

Actually, better interpretation:

- For x < -1: Graph is decreasing overall — no relative extrema.
- For x > -1: Graph rises from -∞ to a peak at about x = 0, y ≈ 0, then decreases.

So:

✔ Relative Maximum: At (0, 0) — highest point in its neighborhood on the right branch.
✔ No relative minimum — the function goes to -∞ on both sides of the asymptote.

> Note: Some might argue there’s a local min on the left if the graph curves up slightly, but visually, it’s monotonic decreasing to the asymptote. So safest answer: only one relative max at (0,0).

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## Problems 3 & 4: Use graphing calculator to approximate relative extrema

*(Since we don’t have a physical calculator, we’ll simulate what you’d see on Desmos or TI-84.)*

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Problem 3: y = -x³ + 4x² - 4

This is a cubic with negative leading coefficient → falls to right, rises to left.

Find critical points by taking derivative:

y’ = -3x² + 8x

Set to zero:
-3x² + 8x = 0
x(-3x + 8) = 0 → x = 0, x = 8/3 ≈ 2.67

Evaluate y at these points:

- At x = 0: y = -0 + 0 - 4 = -4
- At x = 8/3 ≈ 2.67:
y = -(8/3)³ + 4*(8/3)² - 4
= -512/27 + 4*(64/9) - 4
= -512/27 + 256/9 - 4
= -512/27 + 768/27 - 108/27
= (768 - 512 - 108)/27 = 148/27 ≈ 5.48

✔ Relative Minimum: at (0, -4)
✔ Relative Maximum: at (8/3, 148/27) ≈ (2.67, 5.48)

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Problem 4: y = x² / (4x + 4)

Simplify: y = x² / [4(x + 1)] → undefined at x = -1 (vertical asymptote)

Take derivative using quotient rule:

Let u = x², v = 4x + 4
u’ = 2x, v’ = 4

y’ = (u’v - uv’) / v²
= [2x(4x+4) - x²(4)] / (4x+4)²
= [8x² + 8x - 4x²] / (4x+4)²
= (4x² + 8x) / (4x+4)²
= 4x(x + 2) / [16(x+1)²]
= x(x + 2) / [4(x+1)²]

Set numerator = 0:
x(x + 2) = 0 → x = 0, x = -2

Check behavior:

- At x = -2: y = (-2)² / (4*(-2)+4) = 4 / (-8+4) = 4/-4 = -1
- At x = 0: y = 0 / 4 = 0

Now test intervals:

- For x < -2: say x = -3 → y’ = (-3)(-1)/[4(-2)²] = (+3)/16 > 0 → increasing
- Between -2 and -1: say x = -1.5 → y’ = (-1.5)(0.5)/[4(0.5)²] = (-0.75)/1 < 0 → decreasing
- Between -1 and 0: say x = -0.5 → y’ = (-0.5)(1.5)/[4(0.5)²] = (-0.75)/1 < 0 → decreasing
- For x > 0: say x = 1 → y’ = (1)(3)/[4(2)²] = 3/16 > 0 → increasing

So:

✔ Relative Maximum: at x = -2, y = -1
✔ Relative Minimum: at x = 0, y = 0

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## Problems 5 & 6: Approximate intervals of increase/decrease from graphs

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Problem 5:

Graph: starts low on left, increases through origin, peaks around x = 4, then levels off.

From visual inspection:

- Increasing: from x = -∞ to x ≈ 4
- Decreasing: from x ≈ 4 to x = ∞

But wait — after x=4, it flattens out, so technically still increasing but very slowly? Or constant?

Actually, looking closely: after x=4, it appears to be slightly decreasing or leveling off.

More precise:

✔ Increasing on: (-∞, 4)
✔ Decreasing on: (4, ∞)

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Problem 6:

Graph is symmetric, looks like a W-shape, with minima at x = -1 and x = 1, and a local max at x = 0.

From left:

- From -∞ to x = -1: decreasing
- From x = -1 to x = 0: increasing
- From x = 0 to x = 1: decreasing
- From x = 1 to ∞: increasing

✔ Increasing on: (-1, 0) and (1, ∞)
✔ Decreasing on: (-∞, -1) and (0, 1)

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## Problems 7 & 8: Use graphing calculator to find intervals of increase/decrease

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Problem 7: y = x⁴ - 2x² - 3

This is a quartic, even function (symmetric about y-axis).

Derivative: y’ = 4x³ - 4x = 4x(x² - 1) = 4x(x - 1)(x + 1)

Critical points: x = -1, 0, 1

Test intervals:

- x < -1: pick x = -2 → y’ = 4(-2)(-3)(-1) = 4*(-2)*(-3)*(-1) = 4*(-6) = -24 < 0 → decreasing
- -1 < x < 0: pick x = -0.5 → y’ = 4(-0.5)(-1.5)(0.5) = 4*(positive) → positive → increasing
- 0 < x < 1: pick x = 0.5 → y’ = 4(0.5)(-0.5)(1.5) = 4*(negative) → decreasing
- x > 1: pick x = 2 → y’ = 4(2)(1)(3) = 24 > 0 → increasing

✔ Increasing on: (-1, 0) and (1, ∞)
✔ Decreasing on: (-∞, -1) and (0, 1)

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Problem 8: y = -2 / (x² - 1)

Note: Undefined at x = ±1 (vertical asymptotes)

Derivative: Use quotient or chain rule.

Let y = -2(x² - 1)^(-1)

y’ = -2 * (-1)(x² - 1)^(-2) * 2x = 4x / (x² - 1)²

Sign of y’ depends on numerator: 4x

Denominator always positive (squared), except undefined at x=±1.

So:

- y’ > 0 when x > 0 (except x=1)
- y’ < 0 when x < 0 (except x=-1)

Intervals:

✔ Increasing on: (0, 1) and (1, ∞)
✔ Decreasing on: (-∞, -1) and (-1, 0)

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## ✔ Final Answers Summary:

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1)
- Relative Max: (-3, -1)
- Relative Min: (2, -4)

2)
- Relative Max: (0, 0)
- No relative min

3)
- Rel. Min: (0, -4)
- Rel. Max: (2.67, 5.48) or exactly (8/3, 148/27)

4)
- Rel. Max: (-2, -1)
- Rel. Min: (0, 0)

5)
- Increasing: (-∞, 4)
- Decreasing: (4, ∞)

6)
- Increasing: (-1, 0) ∪ (1, ∞)
- Decreasing: (-∞, -1) ∪ (0, 1)

7)
- Increasing: (-1, 0) ∪ (1, ∞)
- Decreasing: (-∞, -1) ∪ (0, 1)

8)
- Increasing: (0, 1) ∪ (1, ∞)
- Decreasing: (-∞, -1) ∪ (-1, 0)

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Let me know if you’d like graphs or more detailed steps for any part!