y(x)=x2+81−x.y(x)=x2+81−x.
1) . , .
1−x=0,⇒x=1.1−x=0,⇒x=1.
x=1x=1 :
D(y)=(−∞;1)∪(1;+∞).D(y)=(−∞;1)∪(1;+∞).
2) . :
, x=1x=1 , x=1x=1 - .
3) .
OyOy, x=0x=0:
, OyOy (0;8)(0;8).
OxOx, y=0y=0:
, OxOx .
, x2+8>0x2+8>0 xx. x∈(−∞;1)x∈(−∞;1) y>0y>0( , ), x∈(1;+∞)x∈(1;+∞) y<0y<0 ( , ).
4) , , :
5) . , - .
6) . :
( y′=0y′=0):
: x=−2,x=1,x=4x=−2,x=1,x=4. :
x∈(−∞;−2),(4;+∞)x∈(−∞;−2),(4;+∞) y′<0y′<0, .
x∈(−2;1),(1;4)x∈(−2;1),(1;4) y′>0y′>0, .
x=−2x=−2 - ( , ), x=4x=4 - ( , ).
:
, (−2;4)(−2;4), (4;−8)(4;−8).
7) . :
:
, . , x∈(−∞;1)x∈(−∞;1) y′′>0y″>0, , x∈(1;+∞)x∈(1;+∞) y′′<0y″<0, .
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8) , .
, .
y=kx+by=kx+b. k,bk,b :
, y=−x−1y=−x−1.
9) . , .
y(−5)=5.5;y(2)=−12;y(7)=−9.5.y(−5)=5.5;y(2)=−12;y(7)=−9.5.
10) , x=1x=1(), y=−x−1y=−x−1 () ( , , ):
4: , ( , )
3.23. , , . a . ?
. x y. S = xy. y - , . 2x + y = a. y = a - 2x S = x(a - 2x),
0 ≤ x ≤ a/2 ( ). S ' = a - 4x, a - 4x = 0 x = a/4,
y = a - 2×a/4 =a/2. x = a/4 - , , . xa/4 S ' > 0, x >a/4 S '< 0, , x=a/4 S . S(a/4) = a/4(a - a/2) = a2 /8 (. ). S [0, a/2] S(0) S(a/2) , . , y = 2x.
3.24. V=16p ≈ 50 3. ( R ), ?
. S = 2pR(R+). V = pR2 Þ = V/pR2 =16p/ pR2 = 16/ R2. , S(R) = 2p(R2+16/R). :
S ' (R) = 2p(2R- 16/R2) = 4p (R- 8/R2). S '(R) = 0 R3 = 8, ,
R = 2, = 16/4 = 4.
3.22. f(x) = 2x3 - 15x2+ 36x - 14.
. f '(x) = 6x2 - 30x +36 = 6(x -2)(x - 3), x1 = 2 x2 = 3. . x1 = 2 , . x2 = 3 , x2 = 3 .
x1 = 2 x2 = 3, : f(2) = 14 f(3) = 13.
3.23. , , . a . ?
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. x y. S = xy. y - , . 2x + y = a. y = a - 2x S = x(a - 2x),
0 ≤ x ≤ a/2 ( ). S ' = a - 4x, a - 4x = 0 x = a/4,
y = a - 2×a/4 =a/2. x = a/4 - , , . xa/4 S ' > 0, x >a/4 S '< 0, , x=a/4 S . S(a/4) = a/4(a - a/2) = a2 /8 (. ). S [0, a/2] S(0) S(a/2) , . , y = 2x.
3.24. V=16p ≈ 50 3. ( R ), ?
. S = 2pR(R+). V = pR2 Þ = V/pR2 =16p/ pR2 = 16/ R2. , S(R) = 2p(R2+16/R). :
S ' (R) = 2p(2R- 16/R2) = 4p (R- 8/R2). S '(R) = 0 R3 = 8, ,
R = 2, = 16/4 = 4.