The figure shows the graph of f(x). Then the correct...

The figure shows the graph of $f(x)$. Then the correct statement(s) is/are
(A) $\mathrm{f}$ is decreasing in $(-1,1)$
(B) $\mathrm{f}$ has a local minimum at $\mathrm{x}=2$
(C) $\mathrm{f}$ is an odd function
(D) $\mathrm{f}$ has atmost 4 zeros.

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Key Concepts

Monotonicity of Functions

Monotonicity refers to the property of a function of being either entirely non-increasing or non-decreasing on an interval. This property helps to analyze intervals where the function consistently rises or falls, which is essential for understanding behavior such as local extrema.

Local Extrema

Local extrema include local minimums and maximums, points where the function values are lower or higher than those of nearby points, respectively. Identifying these points is important for understanding the overall shape and turning points of a function's graph.

Odd Functions

An odd function is defined by the property f(–x) = –f(x) for all x in its domain. This condition implies symmetry about the origin, an aspect that plays a key role in classifying functions and predicting the behavior of their graphs.

Zeros of Functions

Zeros of a function are the x-values where the function equals zero. The number of zeros gives insight into where the function crosses the x-axis, and constraints on the number of zeros can be related to the function’s degree or structure in certain contexts.

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