Applications of Derivatives
Monotonicity of Integral Functions
Grade 12
Question:
<p>Let \(f(x) = \int_0^x e^t(t - 1)(t - 2) dt\). Then \(f\) strictly decreases in the interval</p>
<p>(a) \((-\infty, -2)\)</p>
<p>(b) \((-2, -1)\)</p>
<p>(c) \([1, 2]\)</p>
<p>(d) \((2, \infty)\)</p>
Step-by-Step Solution
Key Concept: Apply Leibnitz rule to find the derivative of the integral, then determine where $f'(x) < 0$ for the function to be strictly decreasing.
<p><strong>Step 1:</strong> Apply Leibnitz rule: $f'(x) = e^x(x - 1)(x - 2)$</p><p><strong>Step 2:</strong> For $f$ to be strictly decreasing: $f'(x) < 0$</p><p><strong>Step 3:</strong> Analyze the sign of $f'(x) = e^x(x - 1)(x - 2)$:</p><p>- $e^x > 0$ for all $x$</p><p>- $(x - 1)(x - 2)$ changes sign at $x = 1$ and $x = 2$</p><p><strong>Step 4:</strong> Using the number line method: $f'(x) < 0$ when $1 < x < 2$, i.e., $x \in [1, 2]$</p><p><strong>∴ Answer is (c)</strong></p>
Correct Answer: c