Given the function f (x) = x-1-alnx, it is proved that f (x) ≥ 0 is constant if and only if a = 1
If f (x) is derived, f '(x) = 1-A / X can be obtained. When x = a, the minimum value can be obtained. So the minimum value of F (a) is a-1-alna. If the minimum value ≥ 0 is constant, then f (x) ≥ 0 is constant. If f (a) is derived from a, then f' (a) = - LNA can be obtained
RELATED INFORMATIONS
- 1. If the function f (x) = {(2b-1) x + B-1, x > 0, is an increasing function on R, then the value range of function B is {- X & # 178; + (2-B) x, X ≤ 0 According to the meaning of the title {2b-1 > 2, ﹛2-b>0 ﹛b-1≥f(0), How do these three inequalities come from?
- 2. If the function f (x) = x & # 178; + 2 (A-1) x + 2 is an increasing function on [4, + ∞), then the value range of real number a is?
- 3. Let f (x) = x2 + X, x < 0-x2, X ≥ 0, if f (f (a)) ≤ 2, then the value range of real number a is___ .
- 4. If the function y = x2-2x + 3 has a maximum value of 3 and a minimum value of 2 in the interval [0, M], then the value range of M is () A. [1,∞)B. [0,2]C. (-∞,2]D. [1,2]
- 5. Given that the function f (x) = x * x-2x + 3 has a maximum value of 3 and a minimum value of 2 on the closed interval 0 and m, what is the value of M?
- 6. Given the function f (x) = x2 + AlN & nbsp; X. (I) when a = - 2, find the extremum of function f (x); (II) if G (x) = f (x) + 2x is a monotone increasing function on [1, + ∞), find the value range of real number a
- 7. The known function f (x) = {- x ^ 3 + x ^ 2, x = 1 Known function f (x) = {- x ^ 3 + x ^ 2, x = 1 For any given positive real number a, whether there are two points P and Q on the curve y = f (x), such that △ poq is a right triangle with o as the right vertex, and the middle point of the hypotenuse of the triangle is on the y-axis
- 8. The known function f (x) = alnx-1 / 2x ^ 2 + 1 / 2 The known function f (x) = alnx-1 / 2x ^ 2 + 1 / 2 (a belongs to R and a is not equal to zero) 1. Find the monotone interval of F (x) 2. Is there a real number a such that for any x belonging to [1, + infinity], f (x) is less than or equal to zero? If so, the value range of a is obtained
- 9. The known function f (x) = alnx + 1 / X (1) When a > 0, find the monotone interval and extremum of F (x) (2) When a > 0, for any x > 0, there is ax (2-lnx)
- 10. Given the function f (x) = x2 + AlN & nbsp; X. (I) when a = - 2, find the extremum of function f (x); (II) if G (x) = f (x) + 2x is a monotone increasing function on [1, + ∞), find the value range of real number a
- 11. The function f (x) = x-1-alnx (a ∈ R) is known. It is proved that f (x) ≥ 0 is constant if and only if a = 1 ② Necessity F '(x) = 1-ax = x-ax, where x > 0 (i) When a ≤ 0, f '(x) > 0 is constant, so f (x) is an increasing function on (0, + ∞) And f (1) = 0, so when x ∈ (0,1), f (x) < 0, which is contrary to f (x) ≥ 0 A ≤ 0 does not satisfy the problem (II) when a > 0, ∵ x > A, f '(x) > 0, so f (x) is an increasing function on (a, + ∞); When 0 < x < a, f '(x) < 0, so the function f (x) is a decreasing function on (0, a); ∴f(x)≥f(a)=a-a-alna ∵ f (1) = 0, so when a ≠ 1, f (a) < f (1) = 0, which is in contradiction with F (x) ≥ 0 ∴a=1 In the process of proving the necessity above, what is the meaning of "∵ f (1) = 0, so when a ≠ 1, f (a) < f (1) = 0, which is in contradiction with F (x) ≥ 0?" why is there f (a) < f (1) when a ≠ 1?
- 12. The monotone increasing interval of function y = 2x Λ & #178; + x-3 is
- 13. It is known that the odd function f (x) is a decreasing function defined on (- 1,1), and f (1-T) + (1-T & sup2;)
- 14. The function y = f (x) is a decreasing function defined on R and an odd function. Solve the equation f (X & # 179; - x-1) + F (X & # 178; - 1) = 0
- 15. If the function f (x) is defined as an odd function on (- 1,1) and a decreasing function, if f (x-1) + F (1-x & # 178;) < 0, the value range of X is obtained
- 16. Given that the function f (x) = (A-2) x & # 178; + (A-1) x + 3 is even, then the monotone increasing interval of F (x) is?
- 17. If f (x) = (K-3) x & # 178; + (K-2) x + 3 is an even function, then the increasing interval of the function is________
- 18. Given that even function f (x) (x ≠ 0) is monotone on interval (0, + ∞), what is the sum of all x satisfying f (X & # 178; - 2x-1) = f (x + 1)?
- 19. The function f (x) is even and is a decreasing function on (- ∞ 0). Try to compare the size of F (- 7 / 8) and f (2a & # 178; - A + 1)
- 20. If the function f (x) = ax & # 178; + BX + 3A + B defined on [a-1,2a] is even, then a + B =?