Question

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

Options

• f is an increasing function of x

• f is a decreasing function of x

• f' is not a continuous function of x

• f is neither increasing nor decreasing function of x

Answer 1

Let f(x) = `x/sqrt(a^2 + x^2) - (d - x)/sqrt(b^2 + (d - x)^2), x ∈ R` where a, b and d are non-zero real constants. Then f is an increasing function of x.

Explanation:

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

= `x/sqrt(a^2 + x^2) + ((x - d))/sqrt(b^2 + (x - d)^2)`

f'(x) = `(sqrt(a^2 + x^2)  - (x(2x))/(2sqrt(a^2  +  x^2)))/((a^2 + x^2)) + (sqrt(b^2 + (x - d)^2) - ((x - d)2(x  -  d))/(2sqrt(b^2 + (x - d)^2)))/((b^2 + (x - d)^2)`

= `(a^2 + x^2 - x^2)/(a^2 + x^2)^(3//2) + (b^2 + (x - d)^2 - (x - d)^2)/(b^2 + (x - d)^2)^(3//2)`

= `a^2/(a^2 + x^2)^(3//2) + b^2/(b^2 + (x - d)^2)^(3//2) > 0`

`\implies` f'(x) > 0, ∀ x ∈ R

`\implies` f(x) is increasing function.

Hence, f(x) is an increasing function.