AQA A-Level Mathematics Paper 3, June 2025: Question 3

1 mark · Easy difficulty · Short Answer

Identify whether the function f(x) = e^x is increasing or decreasing, and concave or convex.

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Question

Question 3 asks: The function f is defined by f(x) = e^x for x in the set of real numbers. Identify which one of the following statements describes the function f. Four options with tick boxes are provided: Decreasing and concave, Decreasing and convex, Increasing and concave, Increasing and convex. The question is worth 1 mark.

Mark scheme

Show the mark scheme Mark scheme table for Question 3 showing 1 mark (R1) with AO 2.2a. Marking instructions state: 'Ticks the fourth box', with typical solution: 'Increasing and convex'.

How to answer it

Properties of Exponential Functions: Increasing & Convex

📌 What this question tests

Understanding and applying the calculus definitions for increasing/decreasing functions (first derivative, f′(x)) and convex/concave functions (second derivative, f″(x)), specifically applied to the natural exponential function f(x) = eˣ for all x ∈ ℝ.

Question 3 Walkthrough

AQA A-Level Mathematics • Multiple Choice • 1 Mark

✅ Correct Answer

Box 4: Increasing and convex

Mark Scheme Breakdown:
• R1 (AO 2.2a): Ticks the fourth box ( Increasing and convex ) correctly.

💡 Key Knowledge

  • Increasing function: f′(x) ≥ 0 (strictly increasing if f′(x) > 0) across the domain.
  • Decreasing function: f′(x) ≤ 0 across the domain.
  • Convex curve: f″(x) ≥ 0 across the domain (curves upwards like a bowl, ∪).
  • Concave curve: f″(x) ≤ 0 across the domain (curves downwards like a cave or dome, ∩).
  • Exponential derivative: For f(x) = eˣ, the derivatives are f′(x) = eˣ and f″(x) = eˣ.

📐 Step-by-Step Analysis

  1. Determine whether the function is increasing or decreasing:
    Differentiate f(x) = eˣ with respect to x:
    f′(x) = eˣ
    Since eˣ > 0 for all real numbers x ∈ ℝ, the gradient is always positive.
    Therefore, f(x) is increasing.
  2. Determine whether the function is concave or convex:
    Find the second derivative:
    f″(x) = eˣ
    Since eˣ > 0 for all x ∈ ℝ, the second derivative is strictly positive (f″(x) > 0).
    Therefore, the function is convex.
  3. Conclusion:
    The function f(x) = eˣ is both increasing and convex (the 4th option).

🧠 Exam Technique & Visualisation

  • Graphical check: Sketch y = eˣ. Moving from left to right, the curve goes upwards (increasing). Any chord drawn between two points on the curve lies entirely above the curve itself (definition of convex).
  • Memory aid for convexity:
    • Con-cave: looks like the entrance to a cave (curves downwards, f″(x) ≤ 0).
    • Con-vex: curves upwards like a smile or bowl (f″(x) ≥ 0).

❌ Common Errors

  • Mixing up concave and convex: Confusing the second derivative conditions (thinking f″(x) > 0 means concave).
  • Confusing the function with e⁻ˣ: If the question had f(x) = e⁻ˣ, f′(x) = −e⁻ˣ < 0 (decreasing), but f″(x) = e⁻ˣ > 0 (still convex). Always write down the derivatives first!
  • Multiple tick marks: Ticking more than one box invalidates the response unless one is clearly scribbled out.

Topics

Pure Mathematics · F: Exponentials and logarithms · G: Differentiation

Question and mark scheme from the AQA A-Level Mathematics examination, Paper 3, June 2025. QuestionVault is an independent revision resource; questions remain the copyright of the awarding body.