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Let f(x)=x/(sqrt(a^2+x^2))-(d-x)/(sqrt(b...

Let `f(x)=x/(sqrt(a^2+x^2))-(d-x)/(sqrt(b^2+(d-x)^2)),x in R,` where a, b and d are non-zero real constants. Then

A

`f` is an increasing function of x

B

`f'` is not a continuous function of x

C

`f` is a decreasing function of x

D

`f` is neither increasing nor decreasing function of x

Text Solution

Verified by Experts

The correct Answer is:
A
We have,
`f(x) = (x) /((a^(2) + x^(2)) ^(1//2)) - ((d - x))/( (b^(2) + (d -x) ^(2))^(1//2)) `
Differentiating above w.r.t. x, we get
`" " (a^(2) + x ^(2) )^(1//2) -x (1)/(2) (2x ) /((a^(2) + x ^(2))^(1//2))/((b^(2) + (d - x^(2))`
` - (b^(2) + (d - x ^(2)) ^(1//2) (-1) - (d - x) (2(d - x)(-1))/( 2 (b ^(2) + (d - x )^(2))^(1//2)))/( (b ^(2) + (d - x )^(2)))`
[by using quotient rule of derivative]
`= (a ^(2) + x ^(2) - x ^(2))/(( a^(2) + x^(2))^(3//2)) + (b ^(2) + (d - x )^(2)- (d - x )^(2))/ ((b^(2) + (d - x) ^(2)) ^(3//2))`
` = (a ^(2))/( (a ^(2) + x^(2)) ^(3//2)) + (b^(2))/(( b^(2) + (d - x )^(2) ) ^( 3//2)) gt 0`,
`AA x in R`
Hence, `f(x)` is an increasing function of x.
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