Question

The function \(f(x) = x^5 - 5x^4 + 5x^3 - 1\) has:

• A. One minimum and one maximum
• B. Two minima and one maxima
• C. Two minima and two maxima
• D. None of the above

Hint:

Factor \(f'(x)\) completely and study the sign change of \(f'(x)\) around each root.

Answer 1

The Correct Option is A

Step 1: Differentiate: \(f'(x) = 5x^4 - 20x^3 + 15x^2 = 5x^2(x^2 - 4x + 3) = 5x^2(x-1)(x-3)\). The critical points are \(x = 0\) (double root), \(x = 1\), and \(x = 3\).

Step 2: Study the sign of \(f'(x)\) in each interval. Since \(5x^2 \geq 0\) always, the sign is controlled by \((x-1)(x-3)\). For \(x<0\): positive. For \(0<x<1\): positive. For \(1<x<3\): negative. For \(x>3\): positive.

Step 3: At \(x=0\), \(f'(x)\) is positive both just before and just after \(x=0\), so the sign does not change. Hence \(x=0\) is NOT a point of local maximum or minimum; it is a point of inflection with a horizontal tangent.

Step 4: At \(x=1\), \(f'(x)\) changes from positive to negative, so \(x=1\) is a local maximum. At \(x=3\), \(f'(x)\) changes from negative to positive, so \(x=3\) is a local minimum.

Step 5: So \(f(x)\) has exactly one maximum and one minimum. The answer is option (A), \(\boxed{\text{One minimum and one maximum}}\).