## Related questions with answers

Suppose that $y=f(x)$ is a solution of the differential equation $d y / d x=g(x)$. Are the statements in the following Problems true or false? If a statement is true, explain how you know. If a statement is false, give a counterexample.

If $g(x)$ is even, then $f(x)$ is odd.

Solutions

VerifiedNote that even functions are defined as $f(-x) = f(x)$; similarly odd functions are defined as $f(-x) = -f(x)$ for all $x$. Provided that $g(x)$, an even function, is the derivative of $f(x)$, which is an odd function, by the definition of derivatives, the following steps should be done

The goal of this task is to determine if the statement is true or false, Suppose that $y=f(x)$ is a solution to the differential equation

$\dfrac{dy}{dx}=g(x)$

If $g(x)$ is even, $f(x)$ is odd.

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