Question

$f(x) = x^3 – 4.5x^2 - 12x$ has a local maximum at x = ___________(an integer value) in the range x = − 2 to + 2

Answer 1

Correct Answer: 1

Step 1: Find the first derivative of \( f(x) \). The first derivative of \( f(x) \) is: \[ f'(x) = 3x^2 - 9x - 12 \] Step 2: Find critical points. To find the critical points, set \( f'(x) = 0 \): \[ 3x^2 - 9x - 12 = 0 \] Divide through by 3: \[ x^2 - 3x - 4 = 0 \] Factorize the quadratic equation: \[ (x - 4)(x + 1) = 0 \] Thus, the critical points are: \[ x = 4 \quad \text{and} \quad x = -1 \] Step 3: Check the second derivative to determine maxima or minima. The second derivative of \( f(x) \) is: \[ f''(x) = 6x - 9 \] At \( x = -1 \): \[ f''(-1) = 6(-1) - 9 = -15 \quad (\text{negative, indicating a local maximum}). \] At \( x = 4 \): \[ f''(4) = 6(4) - 9 = 15 \quad (\text{positive, indicating a local minimum}). \] Step 4: Verify the range. The local maximum \( x = -1 \) lies within the given range \( x = -2 \ \text{to} \ +2 \).