Questions: For the equation (f(x)=-x^3+3 x^2+2), which of the following questions is NOT true? The relative minimum is at the point ((0,2)) The function is increasing on the interval ((2,6)) The relative maximum is at the point ((2,6)) The function is increasing on the interval ((0,2))

For the equation (f(x)=-x^3+3 x^2+2), which of the following questions is NOT true? The relative minimum is at the point ((0,2)) The function is increasing on the interval ((2,6)) The relative maximum is at the point ((2,6)) The function is increasing on the interval ((0,2))

Transcript text: 15 1 point For the equation $f(x)=-x^{3}+3 x^{2}+2$, which of the following questions is NOT true? The relative minimum is at the point $(0,2)$ The function is increasing on the interval $(2,6)$ The relative maximum is at the point $(2,6)$ The function is increasing on the interval $(0,2)$

Solution

Solution Steps

To determine which statements about the function $ f(x) = -x^3 + 3x^2 + 2 $ are not true, we need to analyze the function's critical points and intervals of increase and decrease. First, find the derivative of the function to identify critical points. Then, use the first derivative test to determine where the function is increasing or decreasing. Finally, evaluate the function at critical points to verify the relative minimum and maximum.

Step 1: Find the Derivative and Critical Points

The function given is $ f(x) = -x^3 + 3x^2 + 2 $. To find the critical points, we first calculate the derivative:

\[ f'(x) = \frac{d}{dx}(-x^3 + 3x^2 + 2) = -3x^2 + 6x \]

Setting the derivative equal to zero to find critical points:

\[ -3x^2 + 6x = 0 \]

Factoring out the common term:

\[ -3x(x - 2) = 0 \]

This gives the critical points $ x = 0 $ and $ x = 2 $.

Step 2: Evaluate the Function at Critical Points

Next, we evaluate the function at the critical points to determine the relative extrema:

At $ x = 0 $: \[ f(0) = -(0)^3 + 3(0)^2 + 2 = 2 \]
At $ x = 2 $: \[ f(2) = -(2)^3 + 3(2)^2 + 2 = -8 + 12 + 2 = 6 \]

Thus, the relative minimum is at $ (0, 2) $ and the relative maximum is at $ (2, 6) $.

Step 3: Determine Intervals of Increase and Decrease

To determine where the function is increasing or decreasing, we analyze the sign of the derivative $ f'(x) = -3x^2 + 6x $ between the critical points:

For $ x $ in the interval $ (0, 2) $, choose a test point, say $ x = 1 $: \[ f'(1) = -3(1)^2 + 6(1) = -3 + 6 = 3 > 0 \] The function is increasing on $ (0, 2) $.
For $ x $ in the interval $ (2, \infty) $, choose a test point, say $ x = 3 $: \[ f'(3) = -3(3)^2 + 6(3) = -27 + 18 = -9 < 0 \] The function is decreasing on $ (2, \infty) $.

Final Answer

Based on the analysis:

The relative minimum is at the point $ (0, 2) $.
The function is increasing on the interval $ (0, 2) $.
The relative maximum is at the point $ (2, 6) $.
The function is not increasing on the interval $ (2, 6) $ as it is actually decreasing.

The statement that is NOT true is: "The function is increasing on the interval $ (2, 6) $".

\[ \boxed{\text{The function is increasing on the interval } (2, 6)} \]

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