In each part, sketch the graph of a continuous function f...

In each part, sketch the graph of a continuous function $f$ with the stated properties. (a) $f$ is concave up on the interval $(-\infty,+\infty)$ and has exactly one relative extremum. (b) $f$ is concave up on the interval $(-\infty,+\infty)$ and has no relative extrema. (c) The function $f$ has exactly two relative extrema on the interval $(-\infty,+\infty),$ and $f(x) \rightarrow+\infty$ as $x \rightarrow+\infty$ (d) The function $f$ has exactly two relative extrema on the interval $(-\infty,+\infty),$ and $f(x) \rightarrow-\infty$ as $x \rightarrow+\infty$

In each part, sketch the graph of a continuous function $f$ with the stated properties.
(a) $f$ is concave up on the interval $(-\infty,+\infty)$ and has exactly one relative extremum.
(b) $f$ is concave up on the interval $(-\infty,+\infty)$ and has no relative extrema.
(c) The function $f$ has exactly two relative extrema on the interval $(-\infty,+\infty),$ and $f(x) \rightarrow+\infty$ as $x \rightarrow+\infty$
(d) The function $f$ has exactly two relative extrema on the interval $(-\infty,+\infty),$ and $f(x) \rightarrow-\infty$ as $x \rightarrow+\infty$

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Key Concepts

Asymptotic Behavior

Asymptotic behavior refers to how a function behaves as the input becomes very large or very small. Specifically, limits at infinity describe the end behavior of the function, such as growing arbitrarily high or low, which helps in sketching the overall trend of the graph.

Relative Extrema

Relative or local extrema are the points where a function reaches local maximum or minimum values. These are points where the function changes direction, and they are identified using the first derivative test or the second derivative test. The number and type of relative extrema affect the overall shape of the graph.

Continuity

Continuity ensures that the function has no breaks, jumps, or holes in its graph. For any value within its domain, small changes in the input produce small changes in the output, which is essential for creating smooth, unbroken graphs.

Concavity

Concavity describes the curvature of the graph. When a function is concave up (has a positive second derivative), its graph curves like an upward facing cup. This property influences the shape of the graph and the behavior of tangent lines and relative extrema.

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