<p>Let <span class="math">\(f(x) = \int_0^x e^{x-y} f'(y) dy - (x^2 - x + 1)e^x\)</span>. Find the number of roots of the equation <span class="math">\(f(x) = 0\)</span>.</p>
Step-by-Step Solution
Key Concept: Use differentiation of the integral equation and properties of exponential functions to reduce to a standard form.
<p><strong>Solution:</strong> Differentiate the functional equation to find <span class="math">$f'(x)$</span>. Apply integration by parts to the integral term. The resulting differential equation, combined with initial conditions, yields a unique solution and hence one root.</p>
Correct Answer: 1