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Sketch the graph of a twice-differentiable function y=f(x) whose derivatives have the following sign patterns below. Also you are given:f(-3)=f(0)=0, f′(-3)=f′(-1)=0, f′′(-2)=0.
| Interval | (-∞,-3) | (-3,-1) | (-1,∞) |
| f′(x) | + | - | + |
| Interval | (-∞,-2) | (-2,∞) |
| f′′(x) | - | + |
To sketch this function, focus on the key features provided by the sign patterns of f′(x) and f′′(x), as well as the given points where f(x)=0, f′(x)=0, and f′′(x)=0. Start by plotting these critical points. Then, use the sign of f′(x) to determine where the function is increasing or decreasing, and the sign of f′′(x) to determine the concavity. Remember that a change in the sign of f′(x) indicates a local extremum, while a change in the sign of f′′(x) indicates an inflection point.
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