Additional Mathematics – Calculus | e-Consult

Differentiate using the quotient rule:

g′(x) = [(2)(x² + 1) − (2x + 1)(2x)] / (x² + 1)²

Simplify numerator:

2(x² + 1) − (2x + 1)(2x) = 2x² + 2 − (4x² + 2x) = –2x² − 2x + 2.

Set g′(x)=0 ⇒ –2x² − 2x + 2 = 0 ⇒ x² + x − 1 = 0.

Solve quadratic:

x = [–1 ± √(1 + 4)]/2 = [–1 ± √5]/2.

Second derivative (or sign test) shows:

• At x = (–1 − √5)/2 ≈ –1.618, g′ changes from positive to negative ⇒ local maximum.
• At x = (–1 + √5)/2 ≈ 0.618, g′ changes from negative to positive ⇒ local minimum.

Coordinates (rounded to three decimal places):