I was taking a test and two true/false type questions were asked.
In one of them, I had to say if there is a function $f(x)$ such that $f(x) = f'(x)$. Of course, $e^x$ is such a function and almost everyone who has taken a calculus course knows this fact well.
In the other question, I had to determine if $f(x) = -f'(x)$ was possible.
I was completely stumped at this one. I had never before encountered a function with such property nor did I know how to approach this problem analytically as I am just a high school student.
My question is: is there an analytical way to determine if such a function exists? By analytical, I mean no guessing allowed and just giving an example won't be enough.
Is this possible? If not, can you give an example of a function with the above property?
Answer
If $f'(x)=f(x)$ and if $g(x)=f(-x)$, then $g'(x)=-f'(-x)=-f(-x)=-g(x)$. Can you take it from here?
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