$$ f $$ is a cubic graph with $$ f(3)=f(-3)=f^{\prime}(3) . f^{\prime}(x) $$ is sketch below. Write down the $$ x $$-coordinates of the stationary points of $$ f $$ and classify the stationary points.

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To determine the $$ x $$-coordinates of the stationary points of the cubic function $$ f $$ and classify them, let's analyze the given information and apply calculus principles. ### Given: 1. $$ f $$ is a **cubic function**. 2. $$ f(3) = f(-3) $$. 3. The derivative $$ f'(x) $$ is provided via a sketch (though not visible here, we'll proceed based on typical cubic behavior). ### Understanding the Cubic Function: A **cubic function** generally has the form: $$ f(x) = ax^3 + bx^2 + cx + d $$ Its derivative is: $$ f'(x) = 3ax^2 + 2bx + c $$ Since $$ f'(x) $$ is a **quadratic function**, it can have **two stationary points** (where $$ f'(x) = 0 $$). ### Given $$ f(3) = f(-3) $$: This symmetry suggests that the function's behavior at $$ x = 3 $$ and $$ x = -3 $$ is related, possibly indicating that these points are stationary points (where the slope is zero). ### Identifying Stationary Points: Assuming $$ f'(x) = 0 $$ at $$ x = 3 $$ and $$ x = -3 $$, we set up: $$ f'(3) = 0 $$ $$ f'(-3) = 0 $$ This implies that $$ x = 3 $$ and $$ x = -3 $$ are the $$ x $$-coordinates of the stationary points. ### Classifying the Stationary Points: To classify these points (i.e., determine whether they are local maxima or minima), we examine the second derivative $$ f''(x) $$: $$ f''(x) = 6ax + 2b $$ 1. **At $$ x = 3 $$:** $$ f''(3) = 18a + 2b $$ - If $$ f''(3) > 0 $$, then $$ x = 3 $$ is a **local minimum**. - If $$ f''(3) < 0 $$, then $$ x = 3 $$ is a **local maximum**. 2. **At $$ x = -3 $$:** $$ f''(-3) = -18a + 2b $$ - If $$ f''(-3) > 0 $$, then $$ x = -3 $$ is a **local minimum**. - If $$ f''(-3) < 0 $$, then $$ x = -3 $$ is a **local maximum**. ### Likely Scenario: Given the typical shape of a cubic function and the symmetry $$ f(3) = f(-3) $$: - **$$ x = 3 $$** is likely a **local minimum**. - **$$ x = -3 $$** is likely a **local maximum**. ### Conclusion: - **Stationary Points:** - $$ x = 3 $$: **Local Minimum** - $$ x = -3 $$: **Local Maximum** ### Visual Representation: While a sketch of $$ f'(x) $$ isn't available here, the analysis aligns with the general behavior of cubic functions having one local maximum and one local minimum. --- **Summary:** - **$$ x $$-coordinates of stationary points:** $$ x = 3 $$ and $$ x = -3 $$. - **Classification:** - $$ x = 3 $$: Local **minimum**. - $$ x = -3 $$: Local **maximum**. The stationary points of $$ f $$ are at $$ x = 3 $$ (local minimum) and $$ x = -3 $$ (local maximum).

is a cubic graph with is
sketch below. Write down the -coordinates of
the stationary points of and classify the
stationary points.

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is a cubic graph with is sketch below. Write down the -coordinates of the stationary points of and classify the stationary points.