i) \( f(x)=-x^{3}+6 x^{2}-9 x+4 \)

Certainly! Let's analyze the function provided: \[ f(x) = -x^{3} + 6x^{2} - 9x + 4 \] This analysis will cover **factoring the polynomial**, **finding its roots**, **determining critical points**, and **discussing the overall behavior of the function**. --- ### 1. **Factoring the Polynomial** First, factor out \(-1\) to simplify the expression: \[ f(x) = -\left(x^{3} - 6x^{2} + 9x - 4\right) \] Now, let's factor the cubic polynomial inside the parentheses. **a. Finding Rational Roots:** The **Rational Root Theorem** suggests that any possible rational roots \( p/q \) are factors of the constant term divided by factors of the leading coefficient. For the polynomial \( x^{3} - 6x^{2} + 9x - 4 \): - **Possible Rational Roots:** \( \pm1, \pm2, \pm4 \) **b. Testing Possible Roots:** Let's test \( x = 1 \): \[ 1^{3} - 6(1)^{2} + 9(1) - 4 = 1 - 6 + 9 - 4 = 0 \] Since \( f(1) = 0 \), \( x = 1 \) is a root. **c. Polynomial Division:** Using **synthetic division** to divide the polynomial by \( (x - 1) \): \[ \begin{array}{r|rrrr} 1 & 1 & -6 & 9 & -4 \\ \hline & 1 & -5 & 4 & 0 \\ \end{array} \] This yields: \[ x^{3} - 6x^{2} + 9x - 4 = (x - 1)(x^{2} - 5x + 4) \] **d. Factoring Further:** Factor the quadratic \( x^{2} - 5x + 4 \): \[ x^{2} - 5x + 4 = (x - 1)(x - 4) \] **e. Complete Factorization:** Putting it all together: \[ f(x) = - (x - 1)^{2}(x - 4) \] --- ### 2. **Finding the Roots of \( f(x) \)** Set \( f(x) = 0 \): \[ - (x - 1)^{2}(x - 4) = 0 \] **Solutions:** - \( (x - 1)^{2} = 0 \) ⟹ \( x = 1 \) (double root) - \( x - 4 = 0 \) ⟹ \( x = 4 \) **Roots:** \[ x = 1 \quad (\text{with multiplicity } 2), \quad x = 4 \] --- ### 3. **Determining Critical Points** To find the critical points, compute the first derivative \( f'(x) \) and set it to zero. **a. First Derivative:** \[ f'(x) = \frac{d}{dx}\left[-x^{3} + 6x^{2} - 9x + 4\right] = -3x^{2} + 12x - 9 \] Factor the derivative: \[ f'(x) = -3(x^{2} - 4x + 3) = -3(x - 1)(x - 3) \] **b. Setting \( f'(x) = 0 \):** \[ -3(x - 1)(x - 3) = 0 \] **Solutions:** - \( x = 1 \) - \( x = 3 \) **c. Identifying Local Extrema Using the Second Derivative:** Compute the second derivative \( f''(x) \): \[ f''(x) = \frac{d}{dx}[-3x^{2} + 12x - 9] = -6x + 12 \] **Evaluate at Critical Points:** - **At \( x = 1 \):** \[ f''(1) = -6(1) + 12 = 6 > 0 \quad \text{(Local Minimum)} \] - **At \( x = 3 \):** \[ f''(3) = -6(3) + 12 = -6 < 0 \quad \text{(Local Maximum)} \] **d. Function Values at Critical Points:** - **At \( x = 1 \):** \[ f(1) = -1 + 6 - 9 + 4 = 0 \] - **At \( x = 3 \):** \[ f(3) = -27 + 54 - 27 + 4 = 4 \] **Critical Points:** - **Local Minimum:** \( (1, 0) \) - **Local Maximum:** \( (3, 4) \) --- ### 4. **Behavior of the Function** - **End Behavior:** - As \( x \to \infty \), \( f(x) \to -\infty \) (since the leading term is \(-x^{3}\)). - As \( x \to -\infty \), \( f(x) \to \infty \). - **Graph Characteristics:** - **Roots:** \((1, 0)\) with multiplicity 2 (the graph touches the x-axis and turns around) and \((4, 0)\). - **Local Extrema:** A local minimum at \( (1, 0) \) and a local maximum at \( (3, 4) \). - **Shape:** The graph has the general shape of a downward-opening cubic function, intersecting the x-axis at the specified roots and turning at the critical points. --- ### 5. **Summary** - **Factorized Form:** \[ f(x) = - (x - 1)^{2}(x - 4) \] - **Roots:** \[ x = 1 \quad (\text{double root}), \quad x = 4 \] - **Critical Points:** - Local Minimum at \( (1, 0) \) - Local Maximum at \( (3, 4) \) - **End Behavior:** - \( f(x) \to -\infty \) as \( x \to \infty \) - \( f(x) \to \infty \) as \( x \to -\infty \) This comprehensive analysis should provide a clear understanding of the function \( f(x) = -x^{3} + 6x^{2} - 9x + 4 \).