10MathQ

Study guides › Level 10 · Calculus

Lowest and highest points

Level 10 · Calculus · for questions like “f(x) = x² − 10x + 3. At what x is f(x) smallest?”

💡 The idea

At a lowest or highest point the curve is flat for a moment: the derivative is 0.

A U-shaped curve with its lowest point

🪜 The steps

  1. Take the derivative.
  2. Set it to 0 and solve.
  3. For the value itself, put that x back into f.

✏️ Worked examples

1. f(x) = x² − 10x + 3: f′(x) = 2x − 10 = 0, so x = 5.

2. The largest value of f(x) = −x² + 6x: f′ = −2x + 6 = 0 gives x = 3, and f(3) = −9 + 18 = 9.

3. Where is f(x) = x³ − 12x flat? 3x² − 12 = 0, so x² = 4: x = 2 and x = −2.

⚠️ Watch out

🎯 Try these

  1. f(x) = x² − 6x + 1: smallest at x = ?
  2. f(x) = −x² + 8x: largest value?
  3. f(x) = x³ − 27x: flat at x = ?

Answers: 1) 3 2) 16 3) 3 and −3

This guide is also in the 10MathQ app: tap “📖 Study guide” after a question.