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So if I have a function

f(x) = 7x-2

the first derivative is

7

which I'm inclined to think that the second derivative exists because

7 = 0x+7

and the second derivative is

0

makes sense, I guess, because the slope never ever changes. But how am I supposed to put that into context with the rest of the info available? There is no concaving up or down in either direction (though I can imagine it being a really straight curve) and if all I knew was the first and second derivatives, I might just think I'm at an inflection point and that there's a horizon somewhere else. This might be a moot point since I'm likely always gonna have the original function at hand in the real world but I was wondering if anyone else had any thoughts on the derivatives of a straight line. Maybe the second derivative doesn't exist?

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2 Answers

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The information you get is when you only have $f''(x)=0$ then you now that $f$ is a linear function, because this is an "if and only if"-relation.

That is: If $f(x)$ is linear, then $f''(x)=0$. And: If $f''(x)=0$, then $f(x)$ is linear.

That reads: $f(x)$ is linear $\leftrightarrow$ $f''(x)=0$

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The zero function, that is $f(x)=0$, is a special case of linear function. Suppose the linear function has the form $f(x)=k*x+d$, then the zero function has $k=0$ and $d=0$.

Of course, also $f''(x)=0$ is true in this case. Nevertheless, keep in mind, that a zero function is a linear function, so the statement that $f''(x)=0$ implies that $f(x)$ is linear holds.

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