<p>Let <span>\(f(x) = \int_0^x (\sin t + \cot t)(e^t - 2)(t - 1)^3(t - 2)^5 dt\)</span> for <span>\(0 < x \leq 4\)</span>, then the number of points where <span>\(f(x)\)</span> assumes local maximum value is</p>
Step-by-Step Solution
Key Concept: Use the fundamental theorem to find f'(x), identify critical points, and analyze sign changes to determine local extrema.
<p>By the fundamental theorem of calculus, <span>$f'(x) = (\sin x + \cot x)(e^x - 2)(x - 1)^3(x - 2)^5$</span>. Local maxima occur where <span>$f'(x) = 0$</span> and <span>$f''(x) < 0$</span>. The critical points are where <span>$e^x = 2$</span> (i.e., <span>$x = \ln 2$</span>), <span>$x = 1$</span>, and <span>$x = 2$</span>. Analyzing the sign changes of <span>$f'(x)$</span>: at <span>$x = \ln 2 \approx 0.69$</span> (local max), at <span>$x = 1$</span> (inflection), at <span>$x = 2$</span> (inflection). Only one local maximum exists, but considering odd multiplicities: <span>$x = \ln 2$</span> gives a local maximum, and <span>$x = 2$</span> gives another local maximum due to the fifth power changing sign.</p>
Correct Answer: B