19. \( f(x)=x^{3}-8 \)

Certainly! Let's analyze the function \( f(x) = x^3 - 8 \) in detail. We'll explore its key properties, including factoring, roots, derivatives, and graph behavior. ## 1. **Factoring the Polynomial** The given function is a cubic polynomial. We can factor it using the **difference of cubes** formula: \[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \] Here, \( a = x \) and \( b = 2 \) (since \( 2^3 = 8 \)). Applying the formula: \[ f(x) = x^3 - 8 = (x - 2)(x^2 + 2x + 4) \] ## 2. **Finding the Roots** To find the roots of \( f(x) = 0 \), solve: \[ x^3 - 8 = 0 \quad \Rightarrow \quad x^3 = 8 \quad \Rightarrow \quad x = 2 \] So, the real root is \( x = 2 \). To find the **complex roots**, we can solve the quadratic factor: \[ x^2 + 2x + 4 = 0 \] Using the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-2 \pm \sqrt{4 - 16}}{2} = \frac{-2 \pm \sqrt{-12}}{2} = -1 \pm i\sqrt{3} \] **Summary of Roots:** - **Real Root:** \( x = 2 \) - **Complex Roots:** \( x = -1 + i\sqrt{3} \) and \( x = -1 - i\sqrt{3} \) ## 3. **Derivatives and Critical Points** ### First Derivative \( f'(x) \) \[ f'(x) = \frac{d}{dx}(x^3 - 8) = 3x^2 \] **Analysis:** - \( f'(x) = 3x^2 \geq 0 \) for all real \( x \). - The function is **monotonically increasing** since the derivative is non-negative everywhere. - **Stationary Point:** At \( x = 0 \), \( f'(0) = 0 \). However, since the derivative does not change sign around this point, it's an inflection point rather than a local maximum or minimum. ### Second Derivative \( f''(x) \) \[ f''(x) = \frac{d}{dx}(3x^2) = 6x \] **Analysis:** - \( f''(x) = 6x \) - Concave **up** when \( x > 0 \), and concave **down** when \( x < 0 \). - **Inflection Point:** At \( x = 0 \), where the concavity changes. ## 4. **Graph Behavior** ### General Shape - **Type:** Cubic polynomial. - **Leading Term:** \( x^3 \) signifies that as \( x \to \infty \), \( f(x) \to \infty \), and as \( x \to -\infty \), \( f(x) \to -\infty \). - **Monotonicity:** Since \( f'(x) \geq 0 \) everywhere, the function is always increasing, albeit with a horizontal tangent at \( x = 0 \). ### Key Points - **Y-intercept:** Set \( x = 0 \): \[ f(0) = 0^3 - 8 = -8 \] - **X-intercept:** At \( x = 2 \), as previously determined. - **Inflection Point:** At \( x = 0 \), where the concavity changes. ### Sketch of the Graph 1. **Start at \( (-\infty, -\infty) \).** 2. **Pass through the inflection point \( (0, -8) \).** 3. **Rise smoothly to pass through \( (2, 0) \).** 4. **Continue to \( (\infty, \infty) \).**  *(Note: For an accurate graph, consider using graphing software or tools.)* ## 5. **Summary of Properties** - **Function:** \( f(x) = x^3 - 8 \) - **Factored Form:** \( (x - 2)(x^2 + 2x + 4) \) - **Roots:** \( x = 2 \) (real), \( x = -1 \pm i\sqrt{3} \) (complex) - **First Derivative:** \( f'(x) = 3x^2 \) (always non-negative) - **Second Derivative:** \( f''(x) = 6x \) (determines concavity) - **Behavior:** Always increasing with one real root and two complex roots, passing through \( (0, -8) \) and \( (2, 0) \), with an inflection point at \( x = 0 \). If you have a specific question or aspect of \( f(x) = x^3 - 8 \) you'd like to explore further, feel free to ask!