In W3 The Second Derivative quiz:
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).
Hi @Nada2 Great question:
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).
I hope this helps!
Hello @pastorsoto thanks!
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?
Thanks again,
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.
Ok thanks!