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3 puntos
Se considera la funciónf(x)={x220si 0≤x<20a+bxsi 20≤x<5036−x2100si 50≤x≤60f(x)=\begin{cases}\dfrac{x^2}{20} & \text{si } 0\leq x<20 \\ a+bx & \text{si } 20\leq x<50 \\ 36-\dfrac{x^2}{100} & \text{si } 50\leq x\leq60\end{cases}siendo aa y bb números reales.
a
Determine los valores de aa y bb para que la función ff sea continua en todo su dominio.0,75 pts
b
Para a=26a=26 y b=−310b=-\dfrac{3}{10}:
i
Calcule los extremos relativos de ff.0,75 pts
ii
Represente gráficamente ff.0,5 pts
iii
Calcule el área del recinto acotado limitado por el eje OXOX y la gráfica de la función ff.1 pt

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FUNCIÓN DE DISTRIBUCIÓN NORMAL N(0,1)N(0,1)
Distribución normal N(0, 1), con el área de probabilidad hasta z sombreada, el eje vertical en 0 y las marcas menos infinito, 0, z y más infinito.
zz0,000,010,02
0,00,50000,50400,5080
0,10,53980,54380,5478
0,20,57930,58320,5871
0,30,61790,62170,6255
0,40,65540,65910,6628
0,50,69150,69500,6985
0,60,72570,72910,7324
0,70,75800,76110,7642
0,80,78810,79100,7939
0,90,81590,81860,8212
1,00,84130,84380,8461
1,10,86430,86650,8686
1,20,88490,88690,8888
1,30,90320,90490,9066
1,40,91920,92070,9222
1,50,93320,93450,9357
1,60,94520,94630,9474
1,70,95540,95640,9573
1,80,96410,96490,9656
1,90,97130,97190,9726
2,00,97720,97780,9783
2,10,98210,98260,9830
2,20,98610,98640,9868
2,30,98930,98960,9898
2,40,99180,99200,9922
2,50,99380,99400,9941
2,60,99530,99550,9956
2,70,996530,996640,99674
2,80,997440,997520,99760
2,90,998130,998190,99825
3,00,998650,998690,99874
3,10,999030,999060,99910
3,20,999310,999340,99936
3,30,999520,999530,99955
3,40,999660,999680,99969
3,50,999770,999780,99978
3,60,999840,999850,99985
3,70,999890,999900,99990
3,80,999930,999930,99993
3,90,999950,999950,99996
4,00,999970,999970,99997
zz0,030,040,05
0,00,51200,51600,5199
0,10,55170,55570,5596
0,20,59100,59480,5987
0,30,62930,63310,6368
0,40,66640,67000,6736
0,50,70190,70540,7088
0,60,73570,73890,7422
0,70,76730,77040,7734
0,80,79670,79950,8023
0,90,82380,82640,8289
1,00,84850,85080,8531
1,10,87080,87290,8749
1,20,89070,89250,8944
1,30,90820,90990,9115
1,40,92360,92510,9265
1,50,93700,93820,9394
1,60,94840,94950,9505
1,70,95820,95910,9599
1,80,96640,96710,9678
1,90,97320,97380,9744
2,00,97880,97930,9798
2,10,98340,98380,9842
2,20,98710,98750,9878
2,30,99010,99040,9906
2,40,99250,99270,9929
2,50,99430,99450,9946
2,60,99570,99590,9960
2,70,996830,996930,99702
2,80,997670,997740,99781
2,90,998310,998360,99841
3,00,998780,998820,99886
3,10,999130,999160,99918
3,20,999380,999400,99942
3,30,999570,999580,99960
3,40,999700,999710,99972
3,50,999790,999800,99981
3,60,999860,999860,99987
3,70,999900,999910,99991
3,80,999940,999940,99994
3,90,999960,999960,99996
4,00,999970,999970,99997
zz0,060,070,08
0,00,52390,52790,5319
0,10,56360,56750,5714
0,20,60260,60640,6103
0,30,64060,64430,6480
0,40,67720,68080,6844
0,50,71230,71570,7190
0,60,74540,74860,7517
0,70,77640,77940,7823
0,80,80510,80780,8106
0,90,83150,83400,8365
1,00,85540,85770,8599
1,10,87700,87900,8810
1,20,89620,89800,8997
1,30,91310,91470,9162
1,40,92790,92920,9306
1,50,94060,94180,9429
1,60,95150,95250,9535
1,70,96080,96160,9625
1,80,96860,96930,9699
1,90,97500,97560,9761
2,00,98030,98080,9812
2,10,98460,98500,9854
2,20,98810,98840,9887
2,30,99090,99110,9913
2,40,99310,99320,9934
2,50,99480,99490,9951
2,60,99610,99620,9963
2,70,997110,997200,99728
2,80,997880,997950,99801
2,90,998460,998510,99856
3,00,998890,998930,99896
3,10,999210,999240,99926
3,20,999440,999460,99948
3,30,999610,999620,99964
3,40,999730,999740,99975
3,50,999810,999820,99983
3,60,999870,999880,99988
3,70,999920,999920,99992
3,80,999940,999950,99995
3,90,999960,999960,99997
4,00,999980,999980,99998
zz0,09
0,00,5359
0,10,5753
0,20,6141
0,30,6517
0,40,6879
0,50,7224
0,60,7549
0,70,7852
0,80,8133
0,90,8389
1,00,8621
1,10,8830
1,20,9015
1,30,9177
1,40,9319
1,50,9441
1,60,9545
1,70,9633
1,80,9706
1,90,9767
2,00,9817
2,10,9857
2,20,9890
2,30,9916
2,40,9936
2,50,9952
2,60,9964
2,70,99736
2,80,99807
2,90,99861
3,00,99900
3,10,99929
3,20,99950
3,30,99965
3,40,99976
3,50,99983
3,60,99989
3,70,99992
3,80,99995
3,90,99997
4,00,99998
Nota: En el interior de la tabla se da la probabilidad de que la variable aleatoria ZZ, con distribución N(0,1)N(0,1), esté por debajo del valor zz.