Applications of Derivatives
Maxima and Minima
Grade 12
Question:
<p>Locate the position and nature of any turning points of the function <span class="math">y = x^3 - 3x + 2</span>.</p>
<p>(a) Maxima at <span class="math">(1, 0)</span></p>
<p>(b) Minima at <span class="math">(1, 0)</span></p>
<p>(c) Maxima at <span class="math">(-1, 0)</span></p>
<p>(d) None of the above</p>
Step-by-Step Solution
Key Concept: Find critical points by setting first derivative to zero, then analyze sign changes of the derivative to determine whether each critical point is a maximum or minimum.
<p><strong>Step 1:</strong> Differentiate the function:</p><p><span class="math">y = x^3 - 3x + 2</span></p><p><span class="math">\frac{dy}{dx} = 3x^2 - 3</span></p><p><strong>Step 2:</strong> Find stationary points by setting <span class="math">\frac{dy}{dx} = 0</span>:</p><p><span class="math">3x^2 - 3 = 0</span></p><p><span class="math">3(x^2 - 1) = 0</span></p><p><span class="math">3(x-1)(x+1) = 0</span></p><p>So <span class="math">x = 1</span> or <span class="math">x = -1</span></p><p><strong>Step 3:</strong> Use first derivative test to determine nature of critical points and identify that at <span class="math">x = 1</span>, we have a local minimum.</p><p>∴ Answer is (b).</p>
Correct Answer: B