<p>Given functions are <span class="math">f(x) = 5 - |x - 2|</span> and <span class="math">g(x) = |x + 1|</span>, where <span class="math">x \in \mathbb{R}</span>. Find <span class="math">ab</span> where <span class="math">a</span> is the value of <span class="math">x</span> at which maximum of <span class="math">f(x)</span> occurs and <span class="math">b</span> is the value of <span class="math">x</span> at which minimum of <span class="math">g(x)</span> occurs.</p>
Step-by-Step Solution
Key Concept: The maximum of an absolute value expression occurs when the expression inside is zero, and the minimum of an absolute value is 0 at the point where the argument equals zero.
<p><strong>Step 1:</strong> Find the maximum of <span class="math">f(x) = 5 - |x - 2|</span>.</p><p>Since <span class="math">|x - 2| \geq 0</span>, the expression <span class="math">5 - |x - 2|</span> is maximized when <span class="math">|x - 2| = 0</span>, which occurs at <span class="math">x = 2</span>.</p><p>Therefore, <span class="math">a = 2</span>.</p><p><strong>Step 2:</strong> Find the minimum of <span class="math">g(x) = |x + 1|</span>.</p><p>Since <span class="math">|x + 1| \geq 0</span>, the minimum value is 0, which occurs when <span class="math">x + 1 = 0</span>, i.e., <span class="math">x = -1</span>.</p><p>Therefore, <span class="math">b = -1</span>.</p><p><strong>Step 3:</strong> Calculate <span class="math">ab</span>.</p><p><span class="math">ab = 2 \times (-1) = -2</span></p>
Correct Answer: -2