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Question Sketch the function f(x)=−x3−3x2f(x) = -x^3 - 3x^2 and select the correct graph.

Studdy Solution

STEP 1

Assumptions
1. The function to be sketched is f(x)=−x3−3x2 f(x) = -x^3 - 3x^2 .
2. We will analyze the function's behavior by finding its critical points, inflection points, and asymptotes.
3. We will determine the end behavior of the function by examining the leading term.
4. We will plot key points and the general shape of the function based on the analysis.

STEP 2

First, we need to find the critical points of the function, which occur where the derivative is zero or undefined. Let's find the first derivative of the function.
f′(x)=ddx(−x3−3x2) f'(x) = \frac{d}{dx}(-x^3 - 3x^2)

STEP 3

Calculate the derivative using the power rule.
f′(x)=−3x2−6x f'(x) = -3x^2 - 6x

STEP 4

Set the derivative equal to zero to find the critical points.
−3x2−6x=0 -3x^2 - 6x = 0

STEP 5

Factor out the common term.
−3x(x+2)=0 -3x(x + 2) = 0

STEP 6

Set each factor equal to zero and solve for x x .
−3x=0andx+2=0 -3x = 0 \quad \text{and} \quad x + 2 = 0

STEP 7

Solve for x x in each equation.
x=0andx=−2 x = 0 \quad \text{and} \quad x = -2

STEP 8

Now we will find the second derivative to determine concavity and possible inflection points.
f′′(x)=ddx(−3x2−6x) f''(x) = \frac{d}{dx}(-3x^2 - 6x)

STEP 9

Calculate the second derivative using the power rule.
f′′(x)=−6x−6 f''(x) = -6x - 6

STEP 10

Set the second derivative equal to zero to find possible inflection points.
−6x−6=0 -6x - 6 = 0

STEP 11

Solve for x x .
x=−1 x = -1

STEP 12

Determine the end behavior of the function by examining the leading term −x3 -x^3 . As x x approaches infinity, −x3 -x^3 approaches negative infinity, and as x x approaches negative infinity, −x3 -x^3 approaches positive infinity.

STEP 13

Using the critical points, inflection point, and end behavior, we can sketch the graph. The function decreases from positive infinity to x=−2 x = -2 , where it has a local maximum. Then it decreases until x=−1 x = -1 , where it has an inflection point and continues to decrease more steeply. Finally, as x x approaches infinity, the function goes to negative infinity.

STEP 14

Plot the critical points (−2,f(−2)) (-2, f(-2)) and (0,f(0)) (0, f(0)) , the inflection point (−1,f(−1)) (-1, f(-1)) , and draw the curve according to the end behavior and concavity determined by the second derivative.

STEP 15

The sketch will show a cubic function that starts from the top left (positive y y ), has a local maximum at x=−2 x = -2 , changes concavity at x=−1 x = -1 , and continues to decrease steeply to the bottom right (negative y y ).
Since the options a, b, c, and d are not provided in the actual problem, we cannot select one. However, the description in STEP_15 should match the correct graph among the given options.

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