Limits, Continuity & Differentiability
Functions and Derivatives
Grade 12
<p>The minimum value of <i>k</i> (<i>k</i> ∈ ℤ) for which the equation \(e^x = kx^2\) has exactly three real solutions, is</p>
Step-by-Step Solution
Key Concept: Analyze the function $f(x) = \frac{e^x}{x^2}$ using calculus to find critical points and determine when the horizontal line $y = k$ intersects the curve at exactly three points.
<p><strong>Step 1:</strong> Let $f(x) = \frac{e^x}{x^2}$</p><p><strong>Step 2:</strong> Find the derivative: $f'(x) = \frac{(x-2)e^x}{x^3}$</p><p><strong>Step 3:</strong> The function $f(x)$ is increasing on $(-∞, 0) ∪ [2, ∞)$ and decreasing on $(0, 2)$.</p><p><strong>Step 4:</strong> From the graph, for exactly three real solutions, we need $k = \frac{e^2}{4}$</p><p><strong>Step 5:</strong> Since $k ∈ ℤ$ and $\frac{e^2}{4} ≈ 1.847$, the minimum value of $k$ is 2.</p><p>∴ Answer is 2.</p>
Correct Answer: 2