Exams › JEE Advanced › Maths

Let f(x) = 1 + 2x² + 4x⁴ + 6x⁶ +... + 100x¹⁰⁰ be a polynomial in the real variable x. Then f(x) has:

1. neither a maximum nor a minimum
2. only one maximum
3. only one minimum
4. None of these

Correct answer: only one minimum

Solution

f'(x) = 4x + 16x³ + 24x⁵ +... + 10000x⁹⁹ = 4x(1 + 4x² + 6x⁴ +... + 2500x⁹⁸). The factor in parentheses is always positive for real x. So f'(x) = 0 only at x = 0. For x < 0, f'(x) < 0 (decreasing); for x > 0, f'(x) > 0 (increasing). Therefore x = 0 is a minimum. There is exactly one minimum and no maximum.

Related JEE Advanced Maths questions

• What is the proportion between the height of the cone with the largest possible volume that can fit inside a sphere and the sphere's diameter?
• Identify the point on the curve 9y² = x³ where the normal to the curve intersects the coordinate axes at equal distances.
• The normal to the parametric curve defined by x = a(cos θ + sin θ) and y = a(sin θ − cos θ) at any given value of θ:
• How many critical points exist for the function y = f(x) within the interval x ∈ [0, 2], if the function satisfies d²y/dx² = 6x - 4 and has a local minimum value of 5 at x = 1?
• What is the absolute minimum value of y = f(x) on the interval x ∈ [0, 2], given that f(x) satisfies the equation d²y/dx² = 6x - 4 and has a local minimum value of 5 at x = 1?
• The function f(x) = 2 |x| + |x+2| - ||x+2|-2| - |2|x|| has a local minimum or a local maximum at x =

⚔️ Practice JEE Advanced Maths free + battle 1v1 →