(2) y=x+( 1/x)

Derivation of y to x: y'= 1-( 1/x? )，∵x≠0,∴y′= 1-( 1/x？ )< 1.。

(3) y 1=x？ +2x,∴y 1′=2x+2

y2=-x？ +a，y2′=-2x，

∵L is the common tangent of C 1 and C2,

∴2x+2=-2x,x=- 1/2.

When x=- 1/2, y 1'= y2'= 1. ∴ l: y = x+b, and y2=x? +2 times

x？ +2x=x+b，

x？ +2x-x-b=0

1+4b=0

b=- 1/4，

L: y = x- 1/4，

x- 1/4=-x？ +a

x？ +x-(a+ 1/4)=0

1+4(a+ 1/4)=0

a=- 1/2。

(4) f(x)=x？ +ax+b，f′(x)= 2x+a，

g(x)=x？ +cx+d，g′(x)= 2x+c，

∵f′(x)=g′(x),∴a=c。

By f(5)=30,

∴5? +5a+b=30，

5a+b=5

Multiply f(2x+ 1)=4g(x)

(2x+ 1)？ +a(2x+ 1)+b=4(x？ +cx+d)

. . . . .

(5) f(x)=ax？ +x，

f′(x)= 3ax？ + 1=0

When a < 0, x? =- 1/3ax = √ (- 1/3a) has exactly two extreme values, that is, there are three monotonous intervals.

For example, a=-3, x= 1/3 or-1/3,

When x= 1/3, f( 1/3)=2/9, (1/3, 2/9).

When x=- 1/3, f (-1/3) = -2/9 (-1/3,-2/9).

X∈(-∞,-1/3) monotonically decreases,

(-1/3, 1/3) monotonically increases,

(1/3, +∞) monotonically decreases. ′