The function f(x) = 2x^3 - 21x^2 + 60x + 3 is decreasing in the inter…

The correct answer is Option (4) → \((2, 5)\)

A function is decreasing where \(f'(x) \lt 0\).

    \(f'(x) = 6x^2 - 42x + 60\)
    \(f'(x) = 6(x^2 - 7x + 10)\)
    \(f'(x) = 6(x - 2)(x - 5)\)

The sign of f’(x) changes at x = 2 and x = 5:

    \(x \lt 2\): both factors negative, \(f'(x) \gt 0\)
    \(2 \lt x \lt 5\): factors of opposite signs, \(f'(x) \lt 0\)
    \(x \gt 5\): both factors positive, \(f'(x) \gt 0\)

So f is decreasing in (2, 5) and increasing in \((-\infty, 2)\) and in \((5, \infty)\).