Last edited by @suen 2025-01-13T07:35:26Z
10 Likes
就爱用ai写点这种，魔法打败魔法
Dear Peter,
How is everything going? I received your letter and noticed you’ve been struggling with that calculus problem. Don’t worry, it’s a bit tricky, but I’ve worked out the steps for you.
First, let’s look at the case where \bm{a = 1}. To find the monotonic intervals, we need the derivative: \bm{f’(x) = 2x - 7 + \frac{3}{x}}. By setting \bm{f’(x) = 0}, we find the critical points at \bm{x = 0.5} and \bm{x = 3}. Therefore, the function increases on \bm{(0, 0.5)} and \bm{(3, +\infty)}, and decreases between \bm{0.5} and \bm{3}.
As for the second part, where we need \bm{f(x) \ge -6} on the interval \bm{[2, 3e]}, the best approach is to isolate \bm{a}. By rearranging the inequality, we get \bm{a \ge \frac{6x - 3\ln x - 6}{x^2 - x}}. If we define a new function \bm{g(x)} for that right side and find its maximum value within the range \bm{[2, 3e]}, we can determine the range for \bm{a}. After some calculation, you’ll find that \bm{a} must be greater than or equal to the value at \bm{x=2}, which gives us \bm{a \ge 3 - 1.5\ln 2}.
I hope this helps you clear things up! If you have more questions, feel free to ask.
Best regards,
Li Hua
4 Likes