Given the function f (x) = x * 3-x * 2 + X / 2 + 1 / 4, it is proved that there exists x0 belonging to 0 to 1 / 2, such that f (x0) = x0

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• 1. A necessary and sufficient condition for a function f (x) to be differentiable at point x0 is that the left and right derivatives of F (x) at point x0 exist and are equal. / / / / / / / / / / / / / / however, a necessary condition for differentiability is continuity. Is this contrary to the first proposition
• 2. For example, the derivative of continuous function is not necessarily continuous F (x) is differentiable everywhere on (a, b), but there exists x0 ∈ (a, b), so that f '(x0) exists, but f' (x) is discontinuous at x0 Who can give such an example?
• 3. On the problem of function differentiability, is it required that the left and right derivatives are equal and equal to the derivatives at x0? Or is it only necessary that the left and right derivatives are equal? Or if it is a removable discontinuity point, it means that this derivative does not exist, but this derivative is differentiable?
• 4. Why is the derivative of a function at x0 not equal to the limit value of its derivative at x0
• 5. Let f (x) be second-order differentiable on [a, b], and f ″ (x) ＜ 0. It is proved that: ∫ BAF (x) DX ≤ (B-A) f (a + B2)
• 6. If f (x) is differentiable on R and has two real roots, it is proved that its derivative has at least one real root; if f (x) has three real roots, it is proved that its second derivative has better one real root Urgent request
• 7. F (x) has a second derivative at point x = 0, What does f (x) prove to be n-order differentiable (n > = 2) at x = a
• 8. Let the second derivative of H (x) be continuous at x = 0, and H (0) = 0, H '(0) is not equal to 0. It is proved that the curve y = f (x) = (1-cosx) H (x) is continuous at x = 0 It is proved that f (x) must have inflection point at x = 0
• 9. Let f (x) have the second derivative at x = X. it is proved that the limit of [f (X. + H) - 2F (X.) + F (X. - H)] / h ^ 2 at h → 0 is equal to the second derivative of F (X.)
• 10. Let f (x) be continuous on [0,2], have a second derivative in (0,2), and lim (x approaches 1 / 2) = 0,2 ∫ 1,1 / 2F (x) d (x) =It is proved that there is at least one point δ in (0,2) such that f (δ) = 0
• 11. Let the function y = f (x) be defined in a certain interval, where x0 and x0 + Δ X are defined. What do x0 and x0 + Δ x mean
• 12. When h approaches 0, Lim H / F (A-H) - f (a) = 1 / 3, then the derivative of F (a) is
• 13. It is known that the derivative of F (x) at x = x0 is 4, Lim [x → x0] [f (x) - f (x0)] / 2 (x0-x)]=_______
• 14. Given that f (x) is differentiable at x0, then when h approaches 0, f (x0 + H) − f (x0 − h) 2H approaches () A. 12f′(x0)B. f′（x0）C. 2f′（x0）D. 4f′（x0）
• 15. Let f (x) be differentiable at x0, then it is equal to Why Lim f (x + H) - f (X-H) / h = 2F '(x) Why 2F '(x), not 1 / 2 F' (x) F '(x) isn't that 2 in the denominator, and it becomes 1 / 2 after it is put forward?
• 16. Lim x tends to 0 arcsin2x / sin3x
• 17. X tends to 0) Lim [(x + 4) ^ 1 / 2-2] / sin3x
• 18. Let f (x) be differentiable in a neighborhood of x = 0, and lim f '(x) = 1, then f (x) has extremum in x = 0, and the detailed solution is obtained
• 19. Let f (x) be differentiable and find Lim [f (x + △ x)] ^ 2 - [f (x)] ^ 2 △ x → 0 lim [f(x+△x)]^2-[f(x)]^2 △x→0 =lim{[f(x+△x)+f(x)]*[f(x+△x)-f(x)]}/△x Why is it equal to ＝2f(x)lim[f(x+△x)-f(x)]/△x Especially why is it equal to 2F (x) Please give specific reasons,
• 20. Let f (x) be differentiable at point X. = 0, and f (0) = 0 and f '(0) = 3, then the value of LIM (x →∞) [f (x) / x] ()