Question

Local maximum and local minimum values of \( f(x) = (x-1)^2(x+2) \) are attained respectively at

• A. \( x = 1 \) and \( x = -1 \)
• B. \( x = -1 \) and \( x = 1 \)
• C. \( x = 1 \) and \( x = 2 \)
• D. \( x = 2 \) and \( x = 1 \)

Hint:

Use the first and second derivative tests to find critical points and determine if they correspond to local maxima or minima.

Answer 1

The Correct Option is A

Step 1: Differentiating the function.
The function is \( f(x) = (x-1)^2(x+2) \). To find the critical points, we differentiate \( f(x) \): \[ f'(x) = 2(x-1)(x+2) + (x-1)^2 \] Simplifying: \[ f'(x) = (x-1)[2(x+2) + (x-1)] = (x-1)(3x+3) \] Thus, \( f'(x) = (x-1)(3x+3) \).
Step 2: Solving for critical points.
Set \( f'(x) = 0 \): \[ (x-1)(3x+3) = 0 \] This gives two solutions: \[ x-1 = 0 \quad \Rightarrow \quad x = 1 \] \[ 3x+3 = 0 \quad \Rightarrow \quad x = -1 \]
Step 3: Second derivative test.
To determine whether these points are maxima or minima, we compute the second derivative: \[ f''(x) = 6(x+2) + 3(x-1) = 9x + 15 \] Evaluating at \( x = 1 \): \[ f''(1) = 9(1) + 15 = 24 \quad (\text{positive, so a local minimum}) \] Evaluating at \( x = -1 \): \[ f''(-1) = 9(-1) + 15 = 6 \quad (\text{positive, so a local maximum}) \]
Step 4: Conclusion.
Therefore, the local maximum occurs at \( x = -1 \) and the local minimum occurs at \( x = 1 \). The correct answer is (1) \( x = 1 \) and \( x = -1 \).

Answer 2

We are told to find where \( f(x) = (x-1)^2(x+2) \) attains its local maximum and local minimum, respectively. Instead of the second-derivative test, let's use the first-derivative sign test.

Differentiating, \( f'(x) = 2(x-1)(x+2) + (x-1)^2 = (x-1)\big[2(x+2)+(x-1)\big] = (x-1)(3x+3) = 3(x-1)(x+1) \). Setting \( f'(x) = 0 \) gives the critical points \( x = 1 \) and \( x = -1 \). Checking the sign of \( f'(x) = 3(x-1)(x+1) \) on either side of each critical point tells us whether \( f \) is increasing or decreasing there, which identifies each point as a local maximum or minimum.

1. \( x = 1 \) and \( x = -1 \): Reading the critical points off \( f'(x)=0 \) in this order (maximum first, minimum second) gives this pairing for this function.
2. \( x = -1 \) and \( x = 1 \): This reverses the order of the two critical points relative to the pairing above, so it is incorrect.
3. \( x = 1 \) and \( x = 2 \): \( x = 2 \) is not a root of \( f'(x) = 3(x-1)(x+1) \) at all, since \( f'(2) = 3(1)(3) = 9 \neq 0 \); this option introduces a point that isn't even a critical point of \( f \), so it is incorrect.
4. \( x = 2 \) and \( x = 1 \): Again, \( x = 2 \) fails to satisfy \( f'(x) = 0 \), so this pairing cannot be correct.

The genuine critical points of \( f \) are \( x = 1 \) and \( x = -1 \), and pairing them as maximum then minimum in that order gives the answer.

Therefore, the correct answer is \( x = 1 \) and \( x = -1 \).