Whether the statement, there exist a function f such that for all x is true or false.
The given statement is false.
(a)If a function is such that then the graph of f is concave upward.
(b) If a function is such that then the graph of f is concave upward.
It is given that thus graph of f is always below the x-axis.
Also, as the function is a decreasing function.
From concavity test, as the graph of the function is concave upward.
If the graph is concave upward then the graph must cross the x-axis at some point as x approaches infinity.
This implies that the function is positive. So, this is not possible.
Hence, the given statement is false.
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