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JEE Main202429 Jan 2024Evening ShiftMathematicsApplication of DerivativesActual

The function f x = 2 x + 3 x 2 3 , x ∈ R , has

Options

  1. Aexactly one point of local minima and no point of local maxima
  2. Bexactly one point of local maxima and no point of local minima
  3. Cexactly one point of local maxima and exactly one point of local minima
  4. Dexactly two points of local maxima and exactly one point of local minima

Correct answer

C. exactly one point of local maxima and exactly one point of local minima

Step-by-step solution

Given: f ( x ) = 2 x + 3 ( x ) 2 3 ⇒ f ' x = 2 + 2 x - 1 3 ⇒ f ' x = 2 1 + 1 x 1 3 ⇒ f ' x = 2 x 1 3 + 1 x 1 3 The function changes from positive to negative at x = 1 and from negative to positive at x = 0 So, the point of local maxima M is at x = - 1 and the point of local minima m at x = 0

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