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COMEDK2024Evening ShiftMathematicsApplication of DerivativesActual

What is the nature of the function f(x)=x^3-3 x^2+4 x on real numbers?

Options

  1. AIncreasing
  2. BStrictly decreasing
  3. CDecreasing
  4. DConstant

Correct answer

A. Increasing

Step-by-step solution

The function is given by f(x) = x^3 - 3x^2 + 4x . To determine the nature of the function, compute the first derivative with respect to x : f'(x) = d dx (x^3 - 3x^2 + 4x) = 3x^2 - 6x + 4 . Analyze the quadratic expression 3x^2 - 6x + 4 . The discriminant D of this quadratic is given by D = b^2 - 4ac = (-6)^2 - 4(3)(4) = 36 - 48 = -12 . Since the discriminant D 0 ), the quadratic f'(x) = 3x^2 - 6x + 4 is always positive for all real values of x . Because f'(x) > 0 for all x R , the function f(x) is strictly increasi

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