The function f(x)=2+4 x^2+6 x^4+8 x^6 has

The correct answer is option 2: only one minima

f'(x) = 8x + 24x³ + 48x⁵

and f''(x) = 8 + 72x² + 240x⁴ ≥ 0 for all x ∈ R  ∴ f'(x)=0 ⇒ x(8 + 24x² + 48x⁴) = 0

⇒ x = 0.  Since f''(0) > 0

∴ f has a local minima at x = 0. There is no other critical point and f'(x) exists for all x ∈ R.