\( f(x)=x^{3}+12 x^{2}-10 \) a) Give the domain of \( f \) (in interval notation) b) Find the critical numbers of \( f \). c) Determine the intervals on which \( f \) is increasing and decreasing. \( f \) is increasing on: \( f \) is decreasing on: d) Use the First Derivative Test to determine whether each critical point is a relative maximum, minimum, or neither. Relative maxima occur at \( x=\square \) Relative minima occur at \( x=\square \) (Separate multiple answers by commas.) (Separate multiple answers by commas.)

To find the answers to the questions regarding the function \( f(x)=x^{3}+12 x^{2}-10 \), let's break down the functions step by step. **a)** The domain of \( f \): The function \( f(x) \) is a polynomial, and polynomials are defined for all real numbers. Thus, the domain of \( f \) in interval notation is: \[ (-\infty, \infty) \] **b)** Finding the critical numbers of \( f \): First, take the derivative \( f'(x) \): \[ f'(x) = 3x^2 + 24x \] Setting the derivative equal to zero to find critical points: \[ 3x^2 + 24x = 0 \\ 3x(x + 8) = 0 \] This yields \( x = 0 \) and \( x = -8 \) as critical numbers. **c)** To determine the intervals on which \( f \) is increasing and decreasing, we can analyze \( f'(x) = 3x(x + 8) \). The critical points divide the number line into intervals: 1. \( (-\infty, -8) \) 2. \( (-8, 0) \) 3. \( (0, \infty) \) Choosing test points in each interval: - For \( x = -9 \) in \( (-\infty, -8) \): \( f'(-9) = 3(-9)(-1) > 0 \) (increasing) - For \( x = -4 \) in \( (-8, 0) \): \( f'(-4) = 3(-4)(4) < 0 \) (decreasing) - For \( x = 1 \) in \( (0, \infty) \): \( f'(1) = 3(1)(9) > 0 \) (increasing) Thus, \( f \) is increasing on: \( (-\infty, -8) \) and \( (0, \infty) \) \( f \) is decreasing on: \( (-8, 0) \) **d)** Using the First Derivative Test to classify the critical points: - At \( x = -8 \): - \( f' \) changes from \( + \) (increasing) to \( - \) (decreasing), indicating a relative maximum. - At \( x = 0 \): - \( f' \) changes from \( - \) (decreasing) to \( + \) (increasing), indicating a relative minimum. Relative maxima occur at \( x = -8 \) Relative minima occur at \( x = 0 \) Thus, summarizing: - Domain: \( (-\infty, \infty) \) - Critical numbers: \( 0, -8 \) - Increasing on: \( (-\infty, -8), (0, \infty) \) - Decreasing on: \( (-8, 0) \) - Relative maxima at \( x = -8 \) - Relative minima at \( x = 0 \) Historical Background: Did you know that cubic functions like \( f(x) \) have been studied for centuries? Mathematicians noticed that such functions exhibit interesting behaviors, including turning points where they change direction. It was during the Renaissance that the foundations for analyzing these functions through derivatives were laid down, leading to the calculus we know today. Real-World Application: Cubic functions appear frequently in real life, such as in modeling physical phenomena like the displacement of an object in motion under variable acceleration. Understanding the properties of cubic functions helps engineers and scientists predict behavior in everything from construction beams to car design, making it crucial for designing safe and functional structures and vehicles!