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anjali SHARMA Grade: 12
        

 


Please solve the following problem by using Mean Valu theorem only


For any Two real Numbers a and b  | cos a - cos b | <= | a - b | .

8 years ago

Answers : (1)

ronit bhatiya
14 Points
										












I am providing you the step wise solution for your problem:


Step 1. Function cos x is continuous and differentiable for all real numbers. Use the mean value theorem, using 2 real numbers a and b to write



(cos x) ' = [cos a - cos b] / [a - b]


step 2. Take the absolute value of both sides



| (cos x) ' | = | [cos a - cos b] / [a - b] |


(cos x)' = - sin x, hence.

| (cos x) ' | < = 1


step 3. Which gives



| [cos a - cos b] / [a - b] | <= 1


step 4. But



| [cos a - cos b] / [a - b] | = |cos a - cos b| / |a - b|


step 5. When combined with the above gives



|cos a - cos b| / |a - b| <= 1


step 6. Multiply both sides by |a - b| to obtain



|cos a - cos b| <= |a - b|


 


 


Thanks


8 years ago
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