Matched Problem 5 Follow the graphing strategy and analyze the function $$ f(x)=x^{3}-3 x^{2} $$

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**Step 1. Factor and Find the Intercepts** The function is $$ f(x) = x^3 - 3x^2. $$ Factor out the common term: $$ f(x) = x^2(x - 3). $$ - **$$x$$-intercepts:** Set $$ f(x) = 0 $$: $$ x^2(x-3) = 0. $$ This gives $$ x^2 = 0 $$ or $$ x-3=0 $$, so $$ x = 0 \quad (\text{a double root}), \quad x = 3. $$ - **$$y$$-intercept:** Evaluate $$ f(0) $$: $$ f(0) = 0, $$ so the graph passes through $$(0,0)$$. --- **Step 2. Determine the End Behavior** The highest degree term in $$ f(x) $$ is $$ x^3 $$. Hence, as: - $$ x \to \infty $$, $$\quad f(x) \to \infty$$, - $$ x \to -\infty $$, $$\quad f(x) \to -\infty$$. --- **Step 3. Find the Critical Points** Compute the first derivative: $$ f'(x) = \frac{d}{dx}\left(x^3 - 3x^2\right) = 3x^2 - 6x = 3x(x-2). $$ Set $$ f'(x) = 0 $$: $$ 3x(x-2) = 0 \quad \Rightarrow \quad x = 0 \quad \text{or} \quad x = 2. $$ Now, determine the function values at these points: - At $$ x = 0 $$: $$ f(0) = 0^3 - 3(0)^2 = 0. $$ - At $$ x = 2 $$: $$ f(2) = 2^3 - 3(2)^2 = 8 - 12 = -4. $$ **Determining the nature of the critical points:** - For $$ x < 0 $$: Choose $$ x = -1 $$, then $$ f'(-1) = 3(-1)((-1)-2) = 3(-1)(-3) = 9 > 0. $$ - For $$ 0 < x < 2 $$: Choose $$ x = 1 $$, then $$ f'(1) = 3(1)(1-2) = 3(1)(-1) = -3 < 0. $$ - For $$ x > 2 $$: Choose $$ x = 3 $$, then $$ f'(3) = 3(3)(3-2) = 9 > 0. $$ Thus: - At $$ x = 0 $$, $$ f'(x) $$ changes from positive to negative, indicating a **local maximum** at $$(0,0)$$. - At $$ x = 2 $$, $$ f'(x) $$ changes from negative to positive, indicating a **local minimum** at $$(2,-4)$$. --- **Step 4. Determine the Inflection Point and Concavity** Compute the second derivative: $$ f''(x) = \frac{d}{dx}\left(3x^2 - 6x\right) = 6x - 6 = 6(x-1). $$ Set $$ f''(x) = 0 $$ to find the possible inflection point: $$ 6(x-1) = 0 \quad \Rightarrow \quad x = 1. $$ Find the $$ y $$-value at $$ x = 1 $$: $$ f(1) = 1^3 - 3(1)^2 = 1 - 3 = -2. $$ Thus, the inflection point is at $$(1,-2)$$. Determine the concavity: - For $$ x < 1 $$: Pick $$ x = 0 $$; then $$ f''(0) = 6(0-1) = -6 < 0 \quad \text{(concave downward)}. $$ - For $$ x > 1 $$: Pick $$ x = 2 $$; then $$ f''(2) = 6(2-1) = 6 > 0 \quad \text{(concave upward)}. $$ --- **Step 5. Summary of the Graphing Strategy** - **Intercepts:** - $$x$$-intercepts at $$ x=0 $$ (double root) and $$ x=3 $$. - $$y$$-intercept at $$(0,0)$$. - **Critical Points:** - Local maximum at $$(0,0)$$. - Local minimum at $$(2,-4)$$. - **Inflection Point:** - At $$(1,-2)$$. - **Behavior:** - As $$ x \to -\infty $$, $$ f(x) \to -\infty $$ and as $$ x \to \infty $$, $$ f(x) \to \infty $$. - The graph is concave downward for $$ x < 1 $$ and concave upward for $$ x > 1 $$. Using these details, one can sketch a graph of $$ f(x)=x^3-3x^2 $$ that: - Begins with an increasing behavior in $$ (-\infty, 0) $$, - Achieves a local maximum at $$(0,0)$$ (with a horizontal tangent), - Decreases between $$ x=0 $$ and $$ x=2 $$, - Has an inflection point at $$(1,-2)$$, - Reaches a local minimum at $$(2,-4)$$, - Then increases again, crossing the $$ x $$-axis at $$ x=3 $$ and continuing upward as $$ x \to \infty $$. To analyze the function $$ f(x) = x^3 - 3x^2 $$, follow these steps: 1. **Find Intercepts:** - **$$x$$-intercepts:** $$ x = 0 $$ (double root) and $$ x = 3 $$. - **$$y$$-intercept:** $$ (0,0) $$. 2. **Determine End Behavior:** - As $$ x \to \infty $$, $$ f(x) \to \infty $$. - As $$ x \to -\infty $$, $$ f(x) \to -\infty $$. 3. **Find Critical Points:** - **Local Maximum:** $$ (0,0) $$. - **Local Minimum:** $$ (2,-4) $$. 4. **Find Inflection Point:** - **Inflection Point:** $$ (1,-2) $$. 5. **Concavity:** - Concave downward for $$ x < 1 $$. - Concave upward for $$ x > 1 $$. **Summary:** - The graph has a local maximum at $$ (0,0) $$, a local minimum at $$ (2,-4) $$, and an inflection point at $$ (1,-2) $$. - It crosses the $$ x $$-axis at $$ x = 0 $$ and $$ x = 3 $$, and the $$ y $$-axis at $$ (0,0) $$. - The function increases from $$ -\infty $$ to $$ \infty $$, with changes in concavity at $$ x = 1 $$.

Matched Problem 5 Follow the graphing strategy and analyze the function \( f(x)=x^{3}-3 x^{2} \)

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