2020 session of the university exam [1]

Example 1 [2020 session of the senior high school science exam FENGXIANG a known question 12] \ (f '(x) \ ) is \ (f (x) \) of the derivative of the function, and for any real number \ (X \) are satisfied \ (F '(X) = \) \ (E ^ X (2x +. 3) \) \ (+ F (X) \) , \ (F (0) =. 1 \) , then the inequality \ (f (x ) <5e ^ x \) is set to [solution]

$A.(-4，1)$ $B.(-1，4)$ $C.(-\infty，-4)\cup (1，+\infty)$ $D.(-\infty，-1)\cup (4，+\infty)$

Analysis: known equation \ (F '(X) = \) \ (E ^ X (2x +. 3) \) \ (F + (X) \) is deformed into a \ (\ cfrac {f' ( x) -f (X)} E {X ^}. 3 = 2x + \) ,

Order \ (G (X) = \ {F cfrac (X)} E {X} ^ \) , then \ (g '(x) = \ cfrac {f' (x) -f (x)} {e ^ } X \) , then \ (G '(X). 3 = 2x + \) ,

The \ (G (X) = X ^ 2 + 3x + C \) , and because \ (F (0) =. 1 \) , then \ (g (0) = \ cfrac {f (0)} {e ^ 0} = 1 \) , then the clear \ (C = 1 \) ,

Therefore \ (G (X) = X ^ 2 + 3x +. 1 \) , and the inequality \ (f (x) <5e ^ x \) , ie \ (G (X) <. 5 \) , so \ (x ^ 2 . 1 + 3x + <. 5 \) ,

To give \ (X ^ 2 + 3x. 4-<0 \) , solve for \ (-. 4 <X <. 1 \) , was chosen \ (A \) .