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B15 ЕГЭ 2014
   

ДОСРОЧНЫЙ ЕГЭ 28.04.2014

     

1

  Найдите наибольшее значение функции
y = (x − 27 ) e28 − x на отрезке[23;40].

maximize (x-27)*E^(28-x) over [23,40]

2

   y = (x − 4 ) e2x − 7 →  [2;11]

minimize (x-4)*E^(2*x-7) over [2,11]

3

   y = (2 x − 6 ) e13 −4 x →  [2;14].

maximize (2*x-6)*E^(13-4*x) over [2,14]

4

   y = (2 x +15 ) e2 x + 16 →  [−12;−2]

minimize (2*x+15)*E^(2*x+16) over [-12,-2]

     
   

ЭКСТРЕМУМЫ ЛОКАЛЬНЫЕ И ГЛОБАЛЬНЫЕ

   

ТОЧКА МИНИМУМА

Найдите точку минимума функции

  y = x3 − 48 x + 17

local minima x^3-48x+17

local extrema x^3-48x+17

d(x^3-48x+17)/dx>=0

  y = x3 − 3 x2 + 2

local minima x^3-3x^2+2

  y = x3 − 2 x2 + x + 3

local minima x^3-2x^2+x+3

  y = x3 + 5 x2 + 7 x − 5

local minima x^3+5x^2+7x-5

  y = 7 + 12 x x3

local minima -x^3+12x+7

  y = 9 x2 x3

local minima -x^3+9x^2

  y = 5 + 9 x  −x3/3

local minima 5+9x-x^3/3

  y = x3/3 − 9 x  − 7

local minima x^3/3-9x-7

  y = −x/(x2 + 1)

local minima -x/(x^2+1)

  y = −(x2 + 1)/x

local minima -(x^2+1)/x

  y = 25/x + x + 25

local minima 25/x+x+25

  y = (x2 − 6 x + 11 )1/2

local minima sqrt(x^2-6x+11)

  y = (x + 3)2·(x + 5) 1

local minima (x+3)^2(x+5)-1

  y = x3/2 − 3 x + 1

local minima x^(3/2)-3x+1

  y = (2/3)·x3/2 − 2 x + 1

local minima (2/3)x^(3/2)-2x+1

  yx·x1/2 − 3 x + 1

local minima x*x^(1/2)-3x+1

  y = (2/3)·x·x1/2 − 2 x + 1

local minima (2/3)x*(x^(1/2))-2x+1

  y =7x^2 + 2x + 3

local minima 7^(x^2+2x-3)

  y = (x + 16) ex − 16

local minima (x+16)*E^(x-16)

  y = ( 3 −x ) e3 − x

local minima (3-x)*E^(5-x)

  y = ( 3 x2 − 36 x + 36) ex − 36

local minima (3x^2-36x+36)*E^(x-36)

  y = ( x2 − 8 x + 8) e6 − x

local minima (x^2-8x+8)*E^(6-x)

  y = ( x − 2 )2 ex − 5

local minima (x-2)^2*E^(x-5)

  y = ( x + 3 )2 e2 − x

local minima (x+3)^2*E^(2-x)

  y = log5(x2 − 6 x + 12) + 2

local minima log(5,x^2-6 x+12)+2

  y =  2 x − ln ( x + 3 ) + 7

local minima 2x-ln(x+3)+7

  y = 3 x − ln (x + 3)3

local minima 3x-ln((x+3)^3)

  y = 2 x2 − 5 x + ln x − 3

local minima lnx+2x^2-5x-3

  y = (0.5 x)·cos x + sin x из (0;0.5π)

local minima (1/2-x)cosx+sinx for x=0..Pi/2

   

ТОЧКА МАКСИМУМА

Найдите точку максимума функции

  y = x3 − 48 x + 17

local maxima x^3-48x+17

  y = x3 − 3 x2 + 2

local maxima x^3-3x^2+2

  y = x3 + 2 x2 + x + 3

local maxima x^3+2x^2+x+3

  y = x3 − 5 x2 + 7 x − 5

local maxima x^3-5x^2+7x-5

  y = 7 + 12 x x3

local maxima 7+12x-x^3

  y = 9 x2 x3

local maxima 9x^2-x^3

  y = x3/3 − 9 x  − 7

local maxima (1/3)x^3-9x-7

  y = 5 + 9 x  −x3/3

local maxima 5+9x-(1/3)*x^3

  y = (x − 2)2·(x − 4) + 5

local maxima (x-2)^2*(x-4)+5

  y = −(x2 + 289)/x

local maxima -(x^2+289)/x

  y = 16/x + x + 3

local maxima 16/x+x+3

  y = −x/(x2 + 289)

local maxima -x/(x^2+289)

  y = 7 + 6 x − 2·x3/2

local maxima 7+6x-2x^(3/2)

  y = (4 − 4 x x2)1/2

local maxima sqrt(4-4x-x^2)

  y =  −(2/3)·x3/2 + 3 x + 1

local maxima -(2/3)x^(3/2)+3x+1

  y = 7 + 6 x − 2 x·x1/2

local maxima 7+6x-2xsqrtx

  y = −(2/3)·x·x1/2 + 3 x + 1

local maxima -(2/3)sqrt(x^3)+3x+1

  y =116 xx^2

local maxima 11^(6x-x^2)

  y = ( 9 −x ) ex + 9

local maxima (9-x)*E^(x+9)

  y = ( x + 16 ) e16 − x

local maxima (x+16)*E^(16-x)

  y = ( 3 x2 − 36 x + 36) ex + 36

local maxima (3x^2-36x+36)*E^(x-36)

  y = ( x2 − 10 x + 10)  e5 − x

local maxima (x^2-10x+10)*E^(5-x)

  y = ( x − 2 )2 ex − 6

local maxima (x-2)^2*E^(x-6)

  y = ( x + 6 )2 e4 − x

local maxima (x+6)^2*E^(4-x)

  y =   ln ( x + 5 ) − 2 x + 9

local maxima ln(x+5)-2x+9

  y =  ln (x + 5)5 − 5 x

local maxima ln((x+5)^5)-5x

  y =4 x − 4 ln (x + 7)  + 6

local maxima 4*x − 4*ln(x+7)+6

  y = 8 ln (x + 7) − 8 x + 3

local maxima 8ln(x+7)-8x+3

  y = 2 x2 − 13 x + 9 ln x + 8

local maxima 2x^2-13x+9lnx+8

  y = log2(2 +2 x x2)−2

local maxima log(2,2+2x-x^2)-2

  y = (2 − 3)·cos x − 2 sin x + 5 из (0;π/2)

local maxima (2 x-3) cosx-2 sinx+5 over (0,Pi/2)

   

НАИМЕНЬШЕЕ ЗНАЧЕНИЕ

Найдите наименьшее значение функцииу = f (x) на отрезке [a;b]

  y = x3− 27 x  →[0;4]

minimize x^3-27x over [0,4]

  y = x3 − 3 x2 + 2 → [1;4]

minimize x^3-3x^2+2 over [1,4]

  y = x3 − 2 x2 + x + 3→ [1;4]

minimize x^3-2x^2+x+3 over [1,4]

  y = x3 x2 − 40 x + 3  → [0;4]

minimize x^3-x^2-40x+3 over [0,4]

  y = 7 + 12 x x3 →  [-2;2]

minimize 7+12x-x^3 over [-2,2]

  y = 9 x2 x3 →  [-1;5]

minimize 9x^2-x^3 over [-1,6]

  y = x3/3 − 9 x  − 7→  [-3;3]

minimize (1/3)x^3-9x-7 over [-3,3]

  y = (x + 3)2·(x + 5) 1  → [-4;-1]

minimize (x+3)^2(x+5)-1 over [-4,1]

  y = (x2 + 25)/x  → [-10;-1]

minimize (x^2+25)/x over [-10,-1]

  yx + 36/x  → [1;9]

minimize x+36/x over [1,9]

  y = (x2 − 6 x + 13 )1/2

minimize sqrt(x^2-6x+13)

  yx·x1/2 − 3 x + 1 → [1;9]

minimize x*x^(1/2) − 3*x + 1 over [1,9]

  y = (2/3)·x·x1/2 − 3 x + 1 → [1;9]

minimize (2/3)x*x^(1/2)-3x+1 over [1,9]

  y = x3/2 − 3 x + 1 → [1;9]

minimize x^(3/2)-3*x+1 over [1,9]

  y = (2/3)·x3/2 − 3 x + 1  → [1;9]

minimize (2/3)*x^(3/2) − 3*x + 1 over [1,9]

  y =2x^2 + 2x + 5

minimize 2^(x^2+2x+5)

  y = (x − 8) ex − 7  → [6;8]

minimize (x-8)*E^(x-7) over [6,8]

  y = (8 − x) e9 − x  → [3;10]

minimize (8-x)*E^(9-x) over [3,10]

  y = (3 x2 − 36 x + 36) e x − 10  → [8;11]

minimize (3x^2-36x+36)*E^(x-10) over [8,11]

  y = (x2 − 8 x + 8) e2 − x  →  [1;7]

minimize (x^2-8x+8)*E^(2-x) over [1,7]

  y = (x − 2)2 e x − 2 →  [1;4]

minimize (x-2)^2*E^(x-2) over [1,4]

  y = (x + 3)2 e −3 − x → [-5;-1]

minimize (x+3)^2*E^(-3-x) over [-5,-1]

  y =e2x  − 6e2x  + 3→ [1;2]

minimize E^(2*x)-6*E^x+3 over [1,2]

  y =  9 x − ln ( 9 x ) + 3    →  [1/18;5/18]

minimize 9x-ln(9x)+3 over [-1/18,5/18]

  y =  2 x2 − 5 x  + ln x − 3  →  [5/6;7/6]

minimize 2x^2-5x+lnx-3 over [5/6,7/6]

  y =  3 x − ln ( x + 3 )3    →  [−2.5;0]

minimize 3x-ln((x+3)^3) over [-5/2,0]

  y =  4 x − 4 ln ( x + 7 ) + 6    →  [−6.5;0] 

minimize 4x-4ln(x+7)+6 over [-13/2,0]

  y = log3(x2 − 6 x + 10) + 2

minimize log(3,x^2-6x+10)+2

  y = 7 sin x  − 8 x + 9  → [−1.5π;0]

minimize 7sinx-8x+9 over [-3Pi/2,0]

  y =  5 sin x + (24/π)·x + 6 →  [−5π/6;0]

minimize 5sinx+24x/Pi+6 over [-5Pi/6,0]

  y =  13 x − 9 sin x + 9 → [0;0.5π]

minimize 13x-9sinx+9 over [0,Pi/2]

  y = 3 − 5 π/4 + 5 x − 5·21/2·sin x  → [0;0.5π]

minimize 3-5Pi/4+5x-5sqrt2sinx over [0,Pi/2]

  y =  5 cos x − 6 x + 4 → [−1.5π;0] 

minimize 5cosx-6x+4 over [-3Pi/2,0]

  y =  6 cos x + (24/π)·x + 5 → [−2π/3;0]

minimize 6cosx+24x/Pi+5 over [-2Pi/3,0]

  y =  9 cos x + 14 x + 7 → [0;1.5π]

minimize 9cosx+14x+7 over [0,3Pi/2]

  y = 3 + 5 π/4 − 5 x − 5·21/2·cosx→ [0;0.5π]

minimize 3+5Pi/4-5x-5sqrt2*cosx over [0,Pi/2]

  y =  5 tg x − 5 x + 6  → [0;π/4]

minimize 5tgx-5x+6 over [0,Pi/4]

  y =  4 tg x − 4 x − π + 5  →  [−π/4;π/4]

minimize 4tgx-4x-Pi+5 over [-Pi/4,Pi/4]

  y =  4 x − 4 tg x + 12  → [−π/4;0]

minimize 4x-4tgx+12 over [-Pi/4,0]

  y =  2 tg x − 4 x + π − 3 → [−π/3;π/3]

minimize 2tgx-4x+Pi-3 over [-Pi/3,Pi/3]

  y = −14 x + 7 tg x + 35 π + 11 → [−π/3;π/3]

minimize -14x+7tanx+7Pi/2+11 over [-Pi/3,Pi/3]

   

НАИБОЛЬШЕЕ ЗНАЧЕНИЕ

Найдите наибольшее значение функцииу = f (x) на отрезке [a;b]

  y = x3 − 3 x + 4  → [-2;0]

maximize x^3-3x+4 over [-2,0]

   y = x3 − 6 x2  → [-3;3]

maximize x^3-6x^2 over [-3,3]

  y = x3 + 2 x2 + x + 3→ [-4;-1]

maximize x^3+2x^2+x+3 over [-4,-1]

  y = x3 + 2 x2 − 4 x + 4 → [-2;0]

maximize x^3+2x^2-4x+4 over [-2,0]

  y = x3/3 − 9 x  − 7→ [-3;3]

maximize (1/3)x^3-9x-7 over [-3,3]

  y = 7 + 12 x x3 →  [-2;2]

maximize 7+12x-x^3 over [-2,2]

  y = 9 x2 x3 →  [2;10]

maximize 9x^2-x^3 over [2,10]

  y = 5 + 9 x  −x3/3 → [-3;3]

maximize 5+9x-(1/3)x^3 over [-3,3]

  y = (x − 2)2·(x − 4) + 5 → [1;3]

maximize (x-2)^2*(x-4)+5 over [1,3]

  y = x5 − 5 x3 − 20 x→  [−6;1]

maximize x^5-5*x^3-20*x over [-6,1]

  y = 3 x5 −20 x3 −54 x→  [−4;−1]

maximize 3*x^5-20*x^3-54 over [-4,-1]

  y = (x2 + 25)/x → [1;10]

maximize (x^2+25)/x over [1,10]

  yx + 9/x   → [-4;-1]

maximize x+9/x over [-4,-1]

  y = (5 − 4 x x2 )1/2

maximize sqrt(5-4x-x^2)

  y = 3 x − 2·x3/2 → [0;4]

maximize 3x-2x^(3/2) over [0,4]

  y =  −(2/3)·x3/2 + 3 x + 1  → [1;9]

maximize -(2/3)x^(3/2)+3x+1 over [1,9]

  y = 3 x − 2 x·x1/2  → [0;4]

maximize 3x-2xsqrtx over [0,4]

  y =  −(2/3)·x x1/2 + 3 x + 1  → [1;9]

maximize -(2/3)x*x^(1/2)+3x+1 over [1,9]

  y =3−7 − 6xx^2

maximize 3^(-7-6x-x^2)

  y = (8 − x) e x − 7  →  [3;10]

maximize (8-x)*E^(x-7) over [3,10]

  y = (x − 9) e10 − x  →  [-11;11]

maximize (x-9)*E^(10-x) over [-11,11]

  y = (3 x2 − 36 x + 36) e x  →  [-1;4]

maximize (3x^2-36x+36)*E^x over [-1,4]

  y = (x2 − 10 x + 10) e10 − x  →  [5;11]

maximize (x^2-10x+10)*E^(10-x) over [5,11]

  y = (x − 2)2 e x  →  [-5;1]

maximize (x-2)^2*E^x over [-5,1]

  y = (x + 6)2 e −4 − x → [-6;-1]

maximize (x+6)^2*E^(-4-x) over [-6,-1]

  y =   ln ( x + 5 )5 − 5 x  →  [−45;0]

maximize ln((x+5)^5)-5x over[-9/2,0]

  y =  8 ln ( x + 7 ) − 8 x + 3  →  [−65;0]

maximize 8ln(x+7)-8x+3 over[-13/2,0]

  y =   ln ( 11 x ) − 11 x +9  →  [1/22;5/22]

maximize ln(11x)-11x+9 over[1/22,5/22]

  y =  2 x2 − 13 x  + 9 ln x + 8 →  [13/14;15/14]

maximize 2x^2-13x+9lnx+8 over[13/14,15/14]

  y = log5(4 − 2 x x2) + 3

maximize log(5,4-2x-x^2)+3

  y =  10 sin x − (36/π)·x + 7 →  [−5π/6;0]

maximize 10sinx-36x/Pi+7 over [-5Pi/6,0]

  y = 5 sin x − 6 x + 3   → [0;π/2]

maximize 5sinx-6x+3 over [0,Pi/2]

  y = 12 sin x − 6·31/2·x + 31/2·π + 6  → [0;05π]

maximize 12sinx-6sqrt3*x+sqrt3*Pi+6 over [0,Pi/2]

  y = 15 x − 3 sin x + 5 → [−05π;0]

maximize 15x-3sinx+5 over [-Pi/2,0]

  y =  2 cos x − (18/π)·x + 4 →  [−2π/3;0]

maximize 2cosx-18x/Pi+4 over [-2Pi/3,0]

  y = 12 cos x + 6·31/2·x − 2·31/2·π + 6   → [0;05π]

maximize 12cosx+6sqrt3*x-2sqrt3*Pi+6 over [0,Pi/2]

  y =  7 cos x + 16 x − 2  → [−3π/2;0] 

maximize 7cosx+16x-2 over [-3Pi/2,0]

  y = 4 cos x − 20 x + 7 → [0;3π/2]

maximize 4cosx-20x+7 over [0,3Pi/2]

  y =  3 tg x − 3 x + 5  → [−025π;0]

maximize 3tgx-3x+5 over [-Pi/4,0]

  y =  16 tg x − 16 x + 4π − 5 →  [−π/4;π/4] 

maximize 16tan(x)-16x+4Pi-5 over [-Pi/4,Pi/4]

  y =  3 x − 3 tg x − 5 → [0;π/4]

maximize 3x-3tgx-5 over [0,Pi/4]

  y =  14 x − 7 tg x − 35 π+ 11 → [−π/3;π/3]

maximize 14x-7tgx-(7/2)Pi+11 over [-Pi/3,Pi/3]

  y = − 2 tg x + 4 x − π − 3→ [−π/3;π/3]

maximize -2tgx+4x-Pi-3 over [-Pi/3,Pi/3]

 
 
 

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