If the second derivative of a function f(x), i.e., f′′(x) or d^2/dx^2 is equal to 0 for every value of x, what can we say about f(x)?

I understand the case for inferring that f(x) is a line, but could it not also be the case that f(x) = c where c is a constant? The second derivative will also evaluate to 0, so it’s more accurate to conclude that more information (e.g. f’(x)) is needed to infer what’s happening in f(x).

You’re correct in recognizing that the second derivative being zero for every value of x (i.e., ( f’'(x) = 0 )) has specific implications for the nature of the function ( f(x) ).

When ( f’'(x) = 0 ) for all x, it means that the rate of change of the slope of ( f(x) ) is zero. This directly implies that the slope of ( f(x) ) is constant. A function with a constant slope is a linear function, which can be represented as ( f(x) = mx + b ), where ( m ) and ( b ) are constants. Here, ( m ) is the constant slope, and ( b ) is the y-intercept.

Now, regarding your point about ( f(x) = c ) (where ( c ) is a constant), this is actually a special case of the linear function. If ( f(x) = c ), then the function can be rewritten as ( f(x) = 0x + c ). In this format, it becomes clear that it’s a linear function with a slope ( m = 0 ) and a y-intercept ( b = c ). So, indeed, a constant function is a specific type of linear function with a slope of zero.

Therefore, the conclusion that ( f(x) ) is a linear function when ( f’'(x) = 0 ) for all x is correct. This includes both the general linear case with a non-zero slope and the special case of a constant function (where the slope is zero).

So if f(x) was a piecewise constant function like the step function with a restricted domain to an interval where the function is differentiable, we consider f(x) over that interval to be a special case of a line as well but we wouldn’t be able to say much beyond that, correct?

Yes. If ( f(x) ) is a piecewise constant function, like a step function, and we are considering an interval where the function is differentiable, the analysis of the second derivative ( f’'(x) ) would apply to each differentiable segment independently.

In such a segment, if the second derivative ( f’'(x) ) is zero, it implies that within that particular segment, the function behaves like a linear function. Since it’s a constant function over that interval, it’s a special case of a linear function with a slope of zero.