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Lesson 14: Curve Sketching with Derivatives

1. First & Second Derivative Tests

First derivative test:

Second derivative test:

Inflection points: where \(f''(x) = 0\) and concavity changes.

Exercise 1

For \(f(x) = x^3 - 3x,\, f'(x) = 3x² - 3,\, f''(x) = 6x\). Critical points at x=±1.

2. Curve Sketching Steps

  1. Domain & intercepts \((x=0, y=0)\)
  2. Asymptotes (vertical, horizontal)
  3. Critical points \((f'=0)\) → max/min
  4. Concavity & inflection points \((f''=0)\)
  5. Sign chart for \(f',\, f''\)
  6. Sketch: increasing/decreasing, concave up/down

Example: \(f(x) = x^3 - 3x\)

Critical points \(x=±1\), inflection at \(x=0\), local max at \(x=-1\), min at \(x=1\).

Exercise 2

For \((f(x) = x^4 - 4x^2\), which are true? (Select all)

Exercise 3

For \(f(x) = x^3 - 6x^2 + 9x + 1\)
\(f'(x) =\) \(x^2 -\) \(x +\)
Critical points at \(x =\) (, )
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