<p>On [1, <i>e</i>], the least and greatest values of <i>f</i>(<i>x</i>) = <i>x</i><sup>2</sup> ln <i>x</i> are <i>m</i> and <i>M</i> respectively, then [<i>M + m</i>] is: (where [ ] denotes greatest integer function)</p>
Step-by-Step Solution
Key Concept: To find extreme values of a continuous function on a closed interval, evaluate the function at critical points (where f'(x)=0) and endpoints, then compare all values. The least and greatest values among these become m and M respectively.
<p><strong>Step 1: Find the derivative</strong></p><p>f(x) = x² ln x</p><p>f'(x) = 2x ln x + x² · (1/x) = 2x ln x + x = x(2 ln x + 1)</p><p><strong>Step 2: Find critical points in [1, e]</strong></p><p>Setting f'(x) = 0: x(2 ln x + 1) = 0</p><p>Since x > 0 on [1, e], we need: 2 ln x + 1 = 0</p><p>ln x = -1/2</p><p>x = e^(-1/2) = 1/√e ≈ 0.606</p><p>Since 1/√e < 1, this critical point lies outside [1, e]. So there are no critical points in the interior of [1, e].</p><p><strong>Step 3: Check if f is monotonic on [1, e]</strong></p><p>For x ∈ [1, e]: ln x ≥ 0, so 2 ln x + 1 ≥ 1 > 0</p><p>Therefore f'(x) > 0 on [1, e], meaning f is strictly increasing on [1, e].</p><p><strong>Step 4: Evaluate at endpoints</strong></p><p>f(1) = 1² · ln 1 = 1 · 0 = 0</p><p>f(e) = e² · ln e = e² · 1 = e²</p><p><strong>Step 5: Identify m and M</strong></p><p>Since f is strictly increasing on [1, e]:</p><p>m = f(1) = 0 (minimum/least value)</p><p>M = f(e) = e² (maximum/greatest value)</p><p><strong>Step 6: Calculate M + m</strong></p><p>M + m = e² + 0 = e²</p><p>Since e ≈ 2.71828..., we have e² ≈ 7.389</p><p><strong>Step 7: Apply greatest integer function</strong></p><p>[M + m] = [e²] = [7.389...] = 7</p><p><strong>∴ Answer: 7</strong></p>
Correct Answer: 7