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Question no. 6,8 and 10 including this 14th question. Find all values of a for which function f()=8ax-asin6x-7x-sin5x Increases and has no critical points for all x € R ?

Question no. 6,8 and 10 including this 14th question.
Find all values of a for which function f()=8ax-asin6x-7x-sin5x   Increases and has no critical points for all x € R ?

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Grade:12th pass

1 Answers

Raghav
20 Points
3 years ago
Question 6. Differentiating, f’(x) = 2cos 2x – 8(a+1)sin x + (4a^2 + 8a – 14)x
=> 2(2 cos^2 x – 1) – 8(a+1)sin x  + (..)x
=> 4 ((cos^2 x – 2(a+1)cosx + a^2) + 2a – 4 – 2 cosx)
=> 4((cosx – a)^2 + 2(a – 2 – cosx))
Put f’(x) > 0 bcz we have to find that values for which f is always increasing and has no critical points
=> (cosx – a)^2 + 2(a – 2 – cosx) > 0
(cosx – a)^2 is always greater than (or may be equal to zero)
=> a > 2 + cosx
Max. val. of cosx is 1
So, a € (3, infinity).
 
 
It was hard to read this question and solve it like this. Ask the rest separately properly.

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