Definite Integration
Polynomial Divisibility
Grade 12
Question:
<p>If <span class="math">\(f(x), g(x), h(x)\)</span> and <span class="math">\(f'(x)\)</span> are polynomials in <span class="math">\(x\)</span>, and</p><p><span class="math">\(\left(\int_1^x f(x)h(x) dx\right) \left(\int_1^x g(x)f'(x) dx\right) - \left(\int_1^x f(x)f'(x) dx\right) \left(\int_1^x g(x)h(x) dx\right)\)</span></p><p>is divisible by <span class="math">\((x-1)^l\)</span>, find the maximum value of <span class="math">\(l\)</span>.</p>
Step-by-Step Solution
Key Concept: Recognize the expression as a determinant and use properties of polynomial divisibility combined with vanishing of integrals at the lower limit.
<p><strong>Solution:</strong> Apply the Cauchy-Schwarz determinant inequality for integrals. The expression is a determinant of integrals, which evaluates such that at <span class="math">$x=1$</span> all integrals vanish, giving factor <span class="math">$(x-1)^4$</span>.</p>
Correct Answer: 4