Real number x,y satisfy x^2+y^2=1. If the maximum and the minimum value of the expression z=4-y/7-x are m and m respectively, then find the value of (2m+6m)
Answers
Answer:
4.7 approx ... M = 161/208. m = 10/19
Step-by-step explanation:
Given x² + y² = 1
Let x = SinФ and y = Cos Ф.
Now z = (4 - y) / (7 - x)
= (4 - Sin Ф) / (7 - Cos Ф)
Differentiate wrt Ф:
dz/dФ = [ (7 - CosФ) (- cos Ф) - (4 - sin Ф) (sinФ) ] / (7 - Cos Ф)²
= [ - 7 cos Ф + Cos² Ф - 4 SinФ + Sin² Ф ] / (7 - Cos Ф)²
= [ 1 - 4 Sin Ф - 7 CosФ ] / (7 - Cos Ф)²
= [ 1 - 4 y - 7 x ] / (7 - x)²
Equating it to 0 means
4 y + 7 x = 1 ie., y = (1 - 7 x)/4
Substituting in the first equation we get:
x² + (1 - 7 x)² /16 = 1
16 x² + 1 + 49 x² - 14 x = 16
65 x² - 14 x -15 = 0
x = [7 +- √(49+65*15) ] / 65 = 39/65 or -25/65 = -5/13
y = √(1 - x²) = +- 52/65 or +- 12/13
d²Z/dФ² = ..... It should be -ve for Z maximum and +ve for Z minimum.
Z = (4 - y) / (7 - x) will be maximum when y is minimum and x is maximum.
ie., x = 39/65, y = - 52/65
Max Z = M = (4 + 52/65 ) / ( 7 - 39/65) = 322/416 = 161/208
Min Z = m = (4 - 12/13) / (7 + 5/13) = 40/96 = 10/19
Value of 2 M + 6 m = 2 * 161/208 + 6 * 10/19 = 4.7 approx
Answer:
Step-by-step explanation:
Given x² + y² = 1
Let x = SinФ and y = Cos Ф.
Now z = (4 - y) / (7 - x)
= (4 - Sin Ф) / (7 - Cos Ф)
Differentiate wrt Ф:
dz/dФ = [ (7 - CosФ) (- cos Ф) - (4 - sin Ф) (sinФ) ] / (7 - Cos Ф)²
= [ - 7 cos Ф + Cos² Ф - 4 SinФ + Sin² Ф ] / (7 - Cos Ф)²
= [ 1 - 4 Sin Ф - 7 CosФ ] / (7 - Cos Ф)²
= [ 1 - 4 y - 7 x ] / (7 - x)²
Equating it to 0 means
4 y + 7 x = 1 ie., y = (1 - 7 x)/4
Substituting in the first equation we get:
x² + (1 - 7 x)² /16 = 1
16 x² + 1 + 49 x² - 14 x = 16
65 x² - 14 x -15 = 0
x = [7 +- √(49+65*15) ] / 65 = 39/65 or -25/65 = -5/13
y = √(1 - x²) = +- 52/65 or +- 12/13
d²Z/dФ² = ..... It should be -ve for Z maximum and +ve for Z minimum.
Z = (4 - y) / (7 - x) will be maximum when y is minimum and x is maximum.
ie., x = 39/65, y = - 52/65
Max Z = M = (4 + 52/65 ) / ( 7 - 39/65) = 322/416 = 161/208
Min Z = m = (4 - 12/13) / (7 + 5/13) = 40/96 = 10/19
Value of 2 M + 6 m = 2 * 161/208 + 6 * 10/19 = 4.7 approx