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Rolle
f R R
f(2x + y) = 2f(x) + f(y), x, y R
f(0) =0
f(2x) = 2f(x),x R
f(x + y) =f(x) + f(y),x, y R
f 0
R
f
1
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f (0,+) R
f(1) =0, f(x) =x(f(x) 1) x > 0
f(x) =xlnx, x > 0.
a, b (0,+) a
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f R R
f(0) = 1, e2f(x) 2xef(x) =1 x R
f
limx
f(x)
f
(x
)=
1
x2+1
, x R
g R R
g(0) = 0, g(0) =2 (x2 + 1)g(x) + xg(x) =x, x R
h(x) =x2 + 1 (g(x) 1), x R
g
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3
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2
x
ax2 + bx + c, a 0
a,b,c
x2
0
x
ax2 + bx + c = 0, a 0
=b2 4ac
> 0
x1,2 =b
2a
= 0
x1,2 =b
2a
< 0
x
x2 5x + 6 = 0
a = 1, b =
5, c = 6 =
(5)24 1 6 =2524 = 1 >0
x1,2 =(5) 1
2 1 =
5 + 1
2
5 1
2
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2
x1 = 3 x2 =
2
x2 5x + 6 =0
0
2x2 8x + 8 = 0
a = 2, b =
8, c = 8 =
(8)2 4 2 8 =0
x1,2 =(8)2 2
= 2
x1 = 2 x2 =
2
2x2 8x +
8 = 0
0
x2 + x + 8 = 0
a = 1, b =
1, c = 8 =
(1)2
4 1 8 = 31 < 0
x
x = 0
8
x
x2 + 2006x
2007 =0
a = 1, b =
2006, c = 2007
=(2006)2 4 1 2007 =4024036 8028
x2 + 2006x 2007 = 0
x2 + 2007x x 2007 = 0
x2 x + 2007x 2007 = 0
x(x1)+2007(x1) = 0(x 1)(x + 2007) =0
x = 1 x = 2007
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3
> 0
x1,2
ax2+bx+c =a(xx1)(xx2)
= 0
x1,2 =r
ax2 + bx + c =a(x r)2
< 0
x
x2 5x + 6
x = 2, x = 3
x25x+6 =1 (x2) (x3)
2x2 8x + 8
x = 2
, x = 2
2x28x+8 = 2(x2)(x2) = 2(x2)2
x2 + 2yx 3y2
x2
y
x
a = 1, b = 2y, c = 3y2
= b2 4ac =
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3
(2y)2 4 1 (3y2) = 4y2 +12y2 = 16y2 0
x1,2 =
b
2a =
2y16y2
21 =
2y4y
2
x1 =
y, x2 = 3y
x2+2yx3y2 =1 (xy)(x(3y)) =(x y)(x + 3y)
y
a = 3, b = 2x, c = x2
y2
y
c
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3
8
A =
(2x + 3y 1
)2 + 4x + 6y 1
B =(ac + bd)2 + (ad bc)2 C=a7 a6 3a5 + 3a4 + 3a3 3a2 a + 1
D =a2 + b b2 b + (b a)2
A = 7a2 + 10ab + 3b2
B=5
a2 8
ab+ 3
b2
C=2a2 + 5a + 3
D=
a5 +
a+ 1
a + b + c = 0
b2 + 2bc a2
a + b +
a2 + 2ab c2
a + c +
c2 + 2ac b2
b + c
B =(2 + x + x2)2 x3
1
x,y,z,w
x2 + 10y2 + 10z2 + 9w2 = 6(xy + yz+ zw).
0
x, w
A = n3 n2 + n 1 n
A
n
A
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3
(a + b c)2 =a2 + b2 + c2 + 2ab 2ac 2bc
A =x4 + y4 + z4 2x2y2 2y2z2 2z2x2
2
3
f R R
f(2x + y) =2f(x) + f(y),x, y R (1)
f(0) =0
f(m + n) =f(m) + f(n), m, n Z
f
0
f
R.
f
0 f
R.
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9
http://www.mathematica.gr/forum/viewtopic.php?f=109&t=30104
