(1) If the equation f(x)+6a=0 has two equal roots, find the analytical expression of f(x); ?
(2) If the maximum value of the function f(x) is an integer, find the value range of a ...?
Suppose: f (x) = ax 2+bx+c?
What is the solution set of f(x)>-2x (1, 3)? = = & gt？ ax^2+(b+2)x+c>； 0?
= = & gt？ a & lt0,? 1+3=-(b+2)/a，？ 1*3=c/a？ ...( 1)?
1.？ F (x)+6a = ax 2+bx+(c+6a) = 0 has two equal roots?
= = & gt？ Discriminant =0? = = & gt？ b^2-4*a*(c+6a)=0？ ...(2)?
( 1)(2)? = = & gt？ a=- 1/5，b=-6/5，c=-3/5？
f(x)=-(x^2+6x+3)/5？
2.？ ( 1)== >？ b=-4a-2，？ c=3a？
= = & gt？ f(x)=ax^2？ -(4a+2)x？ +3a？
=a*[x？ -(2a+ 1)]^2？ -(4? +a？ + 1/a)？
Is the maximum value an integer? = = & gt？ -(4? +a？ +1/a) is an integer?
= = & gt？ Answer? +1/a is an integer? = = & gt？ a=- 1？
It is known that x=3 is an extreme point of the function f (x) = AlN (1+x)+x2-10x.
(i) find the value of a;
(ii) Find the monotone interval of the function f(x);
(iii) If the line y=b and the image of the function y=f(x) have three intersections, find the value range of b. 。
Sichuan Volume 22, 2008? (finale)
Analysis: The foundation of the topic requires students to have a strong sense of mathematical structure!
(1) because f (x) = AlN (1+x)+x2-10x (x >; - 1)? So f' (x) =+2x- 10, and x=3 is the function f(x)? An extreme point of, so f' (3) =- 16 = 0, so a= 16.
(2) from (1), f' (x) =+2x-10 = (x-1) (x-3), let f' (x) >; 0 has-1
Let f' (x) < 0 have 1
Therefore, f(x) reaches the maximum when x= 1 and the minimum when x=3.
Therefore, the monotonic increasing interval of f(x) is (-1, 1) and (3,+∞), and the monotonic decreasing interval is