Question

If \( f(x) = x^5 - 5x^3 + 5x^2 - 1 \), then the number of critical points of \( f(x) \) is:

• A. 1
• B. 2
• C. 3
• D. 4

Hint:

To find critical points of a polynomial:
1. Calculate the first derivative.
2. Factor the derivative completely.
3. Identify all distinct real roots. Each distinct real root corresponds to a critical point. Don't count repeated roots multiple times as distinct critical points.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The question asks for the number of critical points of the given function \( f(x) \). Critical points are points where the derivative of the function is either zero or undefined. For polynomial functions, the derivative is always defined, so we look for points where \( f'(x) = 0 \).

Step 2: Key Formula or Approach:
1. Find the first derivative of the function, \( f'(x) \).
2. Set \( f'(x) = 0 \) and solve for \( x \). The number of distinct real roots will be the number of critical points.

Step 3: Detailed Explanation:
Given function: \( f(x) = x^5 - 5x^3 + 5x^2 - 1 \).
Differentiate \( f(x) \) with respect to \( x \):
\[ f'(x) = \frac{d}{dx}(x^5 - 5x^3 + 5x^2 - 1) \]
\[ f'(x) = 5x^4 - 15x^2 + 10x \]
To find the critical points, set \( f'(x) = 0 \):
\[ 5x^4 - 15x^2 + 10x = 0 \]
Factor out \( 5x \):
\[ 5x(x^3 - 3x + 2) = 0 \]
One critical point is \( x = 0 \).
Now, we need to find the roots of the cubic equation \( x^3 - 3x + 2 = 0 \).
By inspection, try integer values that divide 2 (\(\pm 1, \pm 2\)):
- If \( x = 1 \): \( (1)^3 - 3(1) + 2 = 1 - 3 + 2 = 0 \). So, \( x = 1 \) is a root. This means \((x-1)\) is a factor.
- Since \( x = 1 \) is a root, \((x-1)\) is a factor. We can perform polynomial division or synthetic division.
Using synthetic division with root 1:
1 | 1 0 -3 2
1 1 -2
-----------------------
1 1 -2 0
The quotient is \( x^2 + x - 2 \).
So, \( x^3 - 3x + 2 = (x-1)(x^2 + x - 2) = 0 \).
Now, factor the quadratic term:
\( x^2 + x - 2 = (x+2)(x-1) \).
So, \( x^3 - 3x + 2 = (x-1)(x+2)(x-1) = (x-1)^2(x+2) = 0 \).
The roots of \( x^3 - 3x + 2 = 0 \) are \( x = 1 \) (a repeated root) and \( x = -2 \).
Combining all roots from \( f'(x) = 0 \):
The distinct real roots are \( x = 0, x = 1, \) and \( x = -2 \).
Thus, there are 3 distinct critical points.

Step 4: Final Answer:
The number of critical points of \( f(x) \) is 3.