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If f(x) =x2/2-2cosx and g(x) =x2/6x-6sinx, show that f is increasing and g is decreasing

If f(x) =x2/2-2cosx and g(x) =x2/6x-6sinx, show that f is increasing and g is decreasing

Grade:12th pass

2 Answers

Arun
25763 Points
3 years ago
Dear Pawan
 
f’(x) = 2x (2 –2 cos x) – 4x sin x / (2 – 2 cosx)^2
 
 = 4x – 4x ( sin x + cos x) / (2- 2 cos)^2
 
hence it is always positive, hence increasing
 
similarly
 
g’(x) = 2x (6x – 6 sin x) – (6 – 6 cosx)x^2 / (6x- 6 sinx)^2
 = it will always be negative. 
Rajendra Vaidya
13 Points
one year ago
f’(x)=2x[1- cos x]+x^2 sin x / 2(1- cos x)^2
as 1-cos x and sin x both are positive function is increasing
g’(x)= 2x[x- sin x] -x^2[1-cos x]/6(x- sin x)^2
and in 0 to 1 2x>x^2 therefore function is decreasing

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