Definite Integration
Integral evaluation with geometric constraints
Grade 12
Question:
<p>Which of the following is true for \(x > 0\)? Where \(g(x) = \int_0^x (3t^2 + 2t + 9) dt\), \(f(x)\) is a decreasing function, \(AB = f(x)i + g(x)j\), and \(AC = g(x)i + f(x)j\) are the two smallest sides of triangle ABC whose circumcentre lies outside the triangle.</p>
<p>(a) \(f(x) > 0, g(x) < 0\)</p>
<p>(b) \(f(x) < 0, g(x) < 0\)</p>
<p>(c) \(f(x) > 0, g(x) > 0\)</p>
<p>(d) \(f(x) < 0, g(x) > 0\)</p>
Step-by-Step Solution
Key Concept: The circumcentre lies outside the triangle if and only if one angle is obtuse. Combined with the constraint that f is decreasing and g is a cubic with positive coefficients, this determines the signs.
<p>Since \(g(x) = \int_0^x (3t^2 + 2t + 9) dt = x^3 + x^2 + 9x > 0\) for \(x > 0\). The circumcentre lying outside the triangle requires the angle at C to be obtuse, which constrains \(f(x)\) to be negative for \(x > 0\).</p>
Correct Answer: d