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JEE Advanced2025MathematicsApplication of DerivativesActual

Let R denote the set of all real numbers. Let f: R R be defined by f(x)= cases 6 x+ x 2 x+ x & if x 0 7 3 & if x=0 cases Then which of the following statements is (are) TRUE?

Options

  1. AThe point x=0 is a point of local maxima of f
  2. BThe point x=0 is a point of local minima of f
  3. CNumber of points of local maxima of f in the interval [ , 6 ] is 3
  4. DNumber of points of local minima of f in the interval [2 , 4 ] is 1

Correct answer

B. The point x=0 is a point of local minima of f

Step-by-step solution

since 3> 7 3 3>f(0) x=0 is local minima ....option (B) is correct Now, aligned & f(x)= 6 x+ x 2 x+ x =1+ 4 x 2 x+ x & f^ (x)= 4[(2 x+ x) 1-x(2+ x)] (2 x+ x)^2 = 4( x-x x) (2 x+ x)^2 & = 4 x( x-x) (2 x+ x)^2 aligned Options (C) & (D) are also correct.

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