Extrema; points of inflexion; optimization practice questions.

Let f(x) = x³ − 3x + 1. Determine the local minimum value of f.

• 3
• −1
• 1
• −3

Answer: −1

f′(x) = 3x² − 3 = 0 gives x = ±1; f″(x) = 6x, so f″(1) = 6 > 0 means a local minimum at x = 1, with value f(1) = 1 − 3 + 1 = −1.

A function satisfies f′(2) = 0 and f″(2) = −5. State what this tells you about the point x = 2.

• f has a local maximum at x = 2
• f has a local minimum at x = 2
• f has a point of inflexion at x = 2
• The test is inconclusive

Answer: f has a local maximum at x = 2

By the second derivative test, f′ = 0 together with f″ < 0 means the curve is concave down at a stationary point, so x = 2 is a local maximum.

A rectangle has a perimeter of 40 cm. Determine the maximum possible area of the rectangle.

• 40 cm²
• 200 cm²
• 100 cm²
• 400 cm²

Answer: 100 cm²

With sides x and 20 − x, the area A = x(20 − x) = 20x − x² has A′ = 20 − 2x = 0 at x = 10, giving a 10 × 10 square with area 100 cm².

For x > 0, let f(x) = x + 4/x. Determine the minimum value of f.

• 2
• 8
• 16
• 4

Answer: 4

f′(x) = 1 − 4/x² = 0 gives x = 2 (for x > 0); f″(x) = 8/x³ > 0 confirms a minimum, and f(2) = 2 + 2 = 4.

A function f has derivative f′(x) = (x − 1)²(x + 2). Determine the nature of the stationary points of f.

• Local minimum at x = −2 only
• Local minimum at x = 1 and local maximum at x = −2
• Local maximum at x = −2 only
• Local extrema at both x = 1 and x = −2

Answer: Local minimum at x = −2 only

f′ changes sign from negative to positive at x = −2 (local minimum), but the squared factor means f′ does not change sign at x = 1, so there is no extremum there.

The function f(x) = x³ satisfies f′(0) = 0. Determine the nature of the stationary point at x = 0.

• Local minimum
• Stationary point of inflexion
• Local maximum
• There is no stationary point at x = 0

Answer: Stationary point of inflexion

f′(x) = 3x² ≥ 0 does not change sign at x = 0, so it is neither a maximum nor a minimum; the concavity changes there, making it a stationary point of inflexion.