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Question:

Let f(x) = x√(a² + x²) - (d - x)√(b² + (d - x)²), x ∈ R, where a, b and d are non-zero real constants. Then :- f is a decreasing function of x; f is neither increasing nor decreasing function of x; f' is not a continuous function of x; f is an increasing function of x

f' is not continuous function of x

f is a decreasing function of x

f is an increasing function of x

f is neither increasing nor decreasing function of x

Solution:

f'(x) = √(a² + x²) - x²/√(a² + x²)(a² + x²) - [-√(b² + (d - x)²) + (d - x)²/√(b² + (d - x)²)(b² + (d - x)²)]
= a²/ (a² + x²)³/² + b²/(b² + (d - x)²)³/²
Hence f(x) is increasing.