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

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Target Exam

CUET

Subject

-- Mathematics - Section A

Chapter

Applications of Derivatives

Question:

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

Options:

only one maxima

only one minima

no maxima and minima

many maxima and minima

Correct Answer:

only one minima

Explanation:

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.