How to prove that a function is bounded in an interval

There are many ways to judge the boundedness, one of the most intuitive is to judge the boundedness according to the monotonicity of the function, and some other rules, such as the boundedness of continuous functions on the closed interval and so on

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1. It is proved that the function y is uniformly continuous in the finite open interval (a, b), then it is bounded in this interval
2. It is proved that if f (x) is differentiable on the interval I and the derivative function on I is bounded, then f (x) is uniformly continuous on I
3. Let f (x) and G (x) be bounded on D. It is proved that f (x) + G (x), f (x) - G (x), f (x) * g (x) are also bounded on D There is another proof that the function y = xsinx is unbounded on (0, + 8), that, + 8 is positive infinity
4. What is the derivative of logx?
5. F (x) = logx / X for the first and second derivatives Dizzy.. Every answer is heavy.
6. What is the derivative of the derivative of F (x) = log (4x logx)
7. What is the derivative of logx ^ 2? What is the result of deriving logx ^ 2
8. It is proved that the derivative of F (x) * LNX + F (x) * (2 / x) = 0
9. Let g (x) = {① e ^ x, X ≤ 0, ② LNX, x > 0, then the solution set of inequality g (x) ≤ 1 about X is () A.（-∞,1] B.(-∞,e] C.[o,e] D.[0,1]
10. Proof: when x > 0, the inequality LNX > = 1-1 / X
11. If f (x) is known to be a differentiable function, then Lim △ x → 0 f ^ 2 (x + △ x) - f ^ 2 (x) / △ x =?
12. Let f '(x) = f (x), f (x) be a differentiable function, f (0) = 1, and f (x) = XF (x) + x ^ 2, find f' (x) and f (x)
13. F (x) is a differentiable function defined on (0, + ∞) and satisfies XF '(x) - f (x) a
14. If e ^ x is a primitive function of F (x), then ∫ XF (x) DX limit
15. If the differentiable function f (x) defined on (0, + ∞) satisfies: X · f '(x) < f (x) and f (1) = 0, then the solution set of F (x) x < 0 is () A. （0，1）B. （0，1）∪（1，+∞）C. （1，+∞）D. ϕ
16. This paper discusses the limit of function f (x) = | x | / X when x tends to 0 Use piecewise function, be specific
17. The derivative function f '(x) of function f (x) is continuous, and f (0) = 0, f' (0) = A. note that the nearest point of curve y = f (x) and P (T, 0) is Q (s, f (s)), and find the limit value Lim s / T (when t tends to 0)
18. Let f (x) be differentiable at x = 0, and f (0) = 0, then find the following limit, where a is not equal to 0 and is a constant lim x→0 [ f(ax)-f(-ax)]/x
19. If the function f (x) is differentiable at x = 0, and f (0) = 0, X tends to 0: then find the limit f (x) / x =?
20. It is known that the two zeros of the function f (x) = AX2 + (B-8) x-a-ab are - 3 and 2 respectively. (I) find f (x); (II) find the range of F (x) when the domain of F (x) is [0,1]