Describe in detail what you need	The graph of f(x) = -2x3 concaves upward in the entire domain. True or False
How many pages	1

Analyzing the Concavity of the Function f(x) = -2x³

In calculus, understanding the concavity of a function is essential for analyzing its behavior and determining local maxima and minima. Let's explore the concavity of the function defined by:

Function Definition

[ f(x) = -2x^3 ]

Step 1: Finding the First Derivative

To analyze the concavity, we first need to find the first derivative of the function:

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

Step 2: Finding the Second Derivative

Next, we calculate the second derivative, which will help us determine the concavity of the function:

[ f''(x) = \frac{d}{dx}(-6x^2) = -12x ]

Step 3: Analyzing the Second Derivative

The second derivative, ( f''(x) = -12x ), will tell us about the concavity of the function:

• The function is concave upwards where the second derivative is positive.
• The function is concave downwards where the second derivative is negative.

Finding Critical Points

To determine where the second derivative changes sign, we set it equal to zero:

[ -12x = 0 \implies x = 0 ]

Step 4: Test the Intervals

Now we need to test the sign of the second derivative in the intervals created by the critical point ( x = 0 ):

1. For ( x < 0 ):

   • Choose ( x = -1 ): [ f''(-1) = -12(-1) = 12 \quad (\text{positive}) ] This indicates that the function is concave upwards on this interval.

2. For ( x > 0 ):

   • Choose ( x = 1 ): [ f''(1) = -12(1) = -12 \quad (\text{negative}) ] This indicates that the function is concave downwards on this interval.

Step 5: Conclusion

Given that:

• The function is concave upwards for ( x < 0 ).
• The function is concave downwards for ( x > 0 ).

We can conclude that the assertion that the graph of ( f(x) = -2x^3 ) concaves upward in the entire domain is False. The function is not concave up everywhere; it exhibits a change in concavity at ( x = 0 ).

Final Answer

False: The graph of ( f(x) = -2x^3 ) does not concave upward in the entire domain; it is concave down for ( x > 0 ).