Question

let m denote the least value of the expression (x^2)+2/ sqrt(x^2 +1)

Answer 1

we have to find the least value of the expression (x² + 2)/√(x² + 1)

solution : let y = (x² + 2)/√(x² + 1)

⇒y = (x² + 1 + 1)/√(x² + 1)

⇒y = (x² + 1)/√(x² + 1) + 1/√(x² + 1)

⇒y = √(x² + 1) + 1/√(x² + 1)

here now we should use an important concept of AM and GM.

i.e., AM ≥ GM

here √(x² + 1) and 1/√(x² + 1) both are positive numbers so we can apply above concept.

AM = [√(x² + 1) + 1/√(x² + 1)]/2

GM = [√(x² + 1) × 1/√(x² + 1)] = 1

so, [√(x² + 1) + 1/√(x² + 1)]/2 ≥ 1

⇒√(x² + 1) + 1/√(x² + 1) ≥ 2

Therefore the least value of the expression (x² + 2)/√(x² + 1), is 2.