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JEE Main2014MathematicsApplication of DerivativesActual

If f x = 3 5 x + 4 5 x - 1 , x ∈ R , then the equation f x = 0 has :

Options

  1. ANo solution
  2. BMore than two solutions
  3. COne solution
  4. DTwo solutions

Correct answer

C. One solution

Step-by-step solution

f ' x = 3 5 x ln 3 5 + 4 5 x ln 4 5 < 0   ∀   x ∈ R ∵ d d x a x = a x ln a Hence, f x is monotonically decreasing. Also, lim x → + ∞ f x → - 1 ∵ lim x → ∞ a x = 0 ,   0 < a < 1 and lim x → - ∞ f x → ∞ ∵ lim x → - ∞ a x = ∞ ,   0 < a < 1 and there is only 1 solution for f x = 0 .

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