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What is the maximum and minimum value of sin x +cos x?
- a)0, -1
- b)1, 0
- c)-1, -√2
- d)√2, –√2
Correct answer is option 'D'. Can you explain this answer?
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What is the maximum and minimum value of sin x +cos x?a)0, -1b)1, 0c)-...
Maximum and Minimum Value of sin x + cos x
Sin x + cos x can be expressed as a single trigonometric function using the identities sin x = cos(90° - x) and cos x = sin(90° - x). Therefore, sin x + cos x can be written as √2 sin(x + 45°).
Maximum Value:
- The maximum value of sin x + cos x occurs when the value of sin(x + 45°) is at its maximum.
- The maximum value of sin(x + 45°) is 1.
- Therefore, the maximum value of sin x + cos x is √2.
Minimum Value:
- The minimum value of sin x + cos x occurs when the value of sin(x + 45°) is at its minimum.
- The minimum value of sin(x + 45°) is -1.
- Therefore, the minimum value of sin x + cos x is -√2.
Therefore, the maximum value of sin x + cos x is √2 and the minimum value is -√2. Hence, option 'D' is the correct answer.
Sin x + cos x can be expressed as a single trigonometric function using the identities sin x = cos(90° - x) and cos x = sin(90° - x). Therefore, sin x + cos x can be written as √2 sin(x + 45°).
Maximum Value:
- The maximum value of sin x + cos x occurs when the value of sin(x + 45°) is at its maximum.
- The maximum value of sin(x + 45°) is 1.
- Therefore, the maximum value of sin x + cos x is √2.
Minimum Value:
- The minimum value of sin x + cos x occurs when the value of sin(x + 45°) is at its minimum.
- The minimum value of sin(x + 45°) is -1.
- Therefore, the minimum value of sin x + cos x is -√2.
Therefore, the maximum value of sin x + cos x is √2 and the minimum value is -√2. Hence, option 'D' is the correct answer.
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What is the maximum and minimum value of sin x +cos x?a)0, -1b)1, 0c)-...
Let y= sin x + cos x
dy/dx=cos x- sin x
For maximum or minimum dy/dx=0
Setting cosx- sin x=0
We get cos x = sin x
x= π/4, 5π/4———-
Whether these correspond to maximum or minimum, can be found from the sign of second derivative.
d^2y/dx^2=-sin x - cos x=-1/√2–1/√2 (for x=π/4) which is negative. Hence x=π/4 corresponds to maximum.For x=5π/4
d^2y/dx^2=-(-1/√2)-(-1/√2)=2/√2 a positive quantity. Hence 5π/4 corresponds to minimum
Maximum value of the function
y= sin π/4 + cos π/4= 2/√2=√2
Minimum value is
Sin(5π/4)+cos (5π/4)=-2/√2=-√2
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What is the maximum and minimum value of sin x +cos x?a)0, -1b)1, 0c)-1, -√2d)√2, –√2Correct answer is option 'D'. Can you explain this answer?
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