Definite Integration
Applications to Equations
Grade 12
Question:
<p>If the function <span>\(\int_0^x f(t) dt = 5\)</span> as <span>\(|x| \to 1\)</span>, then the value of 'a' so that the equation <span>\(2x + \int_0^x f(t) dt = a\)</span> has atleast two roots of opposite signs in <span>\((-1, 1)\)</span> is</p>
<p>(A) \(a \in (0, 1)\)</p>
<p>(B) \(a \in (0, 3)\)</p>
<p>(C) \(a \in (-\infty, 1)\)</p>
<p>(D) \(a \in (3, \infty)\)</p>
Step-by-Step Solution
Key Concept: Use continuity and the intermediate value theorem on the function g(x) to determine when it has roots of opposite signs.
<p>Let <span>$g(x) = 2x + \int_0^x f(t) dt - a$</span>. We need <span>$g(x) = 0$</span> to have two roots of opposite signs. At <span>$x = 0$</span>: <span>$g(0) = -a$</span>. As <span>$x \to 1^-$</span>, <span>$g(x) \to 2 + 5 - a = 7 - a$</span>. As <span>$x \to -1^+$</span>, <span>$g(x) \to -2 + 5 - a = 3 - a$</span>. For two roots of opposite signs, we need <span>$g(0) \neq 0$</span> and appropriate sign changes, which occurs when $a \in (0, 3)$.</p>
Correct Answer: B