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Let the derivative of function f (x) on R be f '(x) and 2F (x) + XF' (x) > x2. The following inequality is always true on R: a.f (x) > 0 B F (x) x D.F (x)
Choose a
analysis:
Not remember g (x) = x ^ 2F (x)
Let g '(x) = x (2f (x) + XF' (x))
=Unique stagnation point of 0 x = 0
When XX * x ^ 2 > 0, G (x) increases
Then min {g (x)} = g (0) = 0
Therefore, there is always g (x) = x ^ 2F (x) > G (0) = 0, X= 0, get f (x) > 0, X= 0
Note that 2F (x) + XF '(x) > x ^ 2 > = 0
Then it is easy to get f (0) > 0
In conclusion, the constant f (x) > 0 holds
Known function f (x)= x2+1 (x≥0) one (x＜0) Then the value range of X satisfying the inequality f (1-x2) > F (2x) is () A. （-1，0） B. （0，1） C. （-1， 2-1） D. （- 2-1， 2-1）
From the meaning of the question, draw the image of function f (x), as shown in the figure:
∵f（1-x2）＞f（2x）
∴
1−x2＞0
2x＜0 or
1−x2＞0
2x≥0
1−x2＞2x
Solution: - 1 ＜ x ＜ 0 or 0 ≤ x ＜
2−1
∴−1＜x＜
2−1
So choose C
It is known that the function f (x) (x ∈ R) satisfies f (1) = 1, and the derivative f '(x) of F (x) is less than 1 2, then inequality f (x2) ＜ x2 2+1 The solution set of 2 is __
Let f (x) = f (x) - 12x, then f ′ (x) = f ′ (x) - 12 ∵ f ′ (x) < 12, ∵ f ′ (x) = f ′ (x) - 12 < 0, that is, the function f (x) decreases monotonically on R, while f (x2) < X22 + 12, that is, f (x2) - X22 < f (1) - 12
Given that the function f (x) is a differentiable function defined on R, and f (- 1) = 2, f '(x) > 2, the solution set of inequality f (x) > 2x + 4 is () A. （-∞，-1） B. （-1，+∞） C. （-1，0） D. （0，+∞）
Let f (x) = f (x) - (2x + 4),
Then f (- 1) = f (- 1) - (- 2 + 4) = 2-2 = 0,
For any x ∈ R, f ′ (x) > 2, so f ′ (x) = f ′ (x) - 2 > 0,
That is, f (x) monotonically increases on R,
Then the solution set of F (x) > 0 is (- 1, + ∞),
That is, the solution set of F (x) > 2x + 4 is (- 1, + ∞)
So choose B
The increasing function f (x) defined on (- 1,1) is known to solve the inequality f (1 + x)
-1
Given that the function f (x) defined on (- 1,1) is both an odd function and a subtractive function, find the inequality f (x ^ 2-2) + F (3-2x)
According to the domain:
-1
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