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Finding maximum and minimum points worksheets Grade 12

Find critical points and classify them with the second derivative.

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How to do it

Set f′(x) = 0 to find the critical points. f″ negative: a maximum. f″ positive: a minimum.

f(x)=−3x3+9x2−3→max at x=2, min at x=0f(x) = -3x^{3} + 9x^{2} - 3\quad \to \quad \text{max at }x = 2,\ \text{min at }x = 0
f′(x)=−9x2+18x=−9x(x−2)f'(x) = -9x^{2} + 18x = -9x(x - 2)set f′(x) = 0
critical points x=0 and x=2\text{critical points }x = 0\ \text{and }x = 2
f′′(x)=−18x+18f''(x) = -18x + 18
f′′(0)=18: min; f′′(2)=−18: maxf''(0) = 18:\ \text{min; }f''(2) = -18:\ \text{max}f″ negative: a max; positive: a min

Example problems

  1. f(x)=x3−9x2+15x+8→f(x) = x^{3} - 9x^{2} + 15x + 8\quad \to \quad
  2. f(x)=3x3+27x2+72x+8→f(x) = 3x^{3} + 27x^{2} + 72x + 8\quad \to \quad
  3. f(x)=−x3−3x2+9x−2→f(x) = -x^{3} - 3x^{2} + 9x - 2\quad \to \quad
  4. f(x)=x3−75x+7→f(x) = x^{3} - 75x + 7\quad \to \quad

Limits and derivatives

  1. Evaluating limits
  2. The power rule for derivatives
  3. The product rule
  4. The quotient rule
  5. The chain rule
  6. Derivatives of trig, exponential and log functions
  7. Equation of a tangent line
  8. Derivatives as rates of change
  9. Finding maximum and minimum points