Question

Let a function be f(x)=0

This will be the equation of the x axis

Integral of f(x) will be the area under the curve f(x) and the x axis. Since f(x) and x axis coincide , area under them will be 0. But antiderivative of 0 is a constant. ( derivative of a constant is 0 so anti derivative of 0 is a constant)

Is there something wrong in the in this statement? Is yes , give reason, if no give reason

Answer 1

Let us understand this question from very basic.

A function f(x) is a function whose value we can say 'y' which varies with x coordinate.
when we write f(x)= 0, it means we put value of y=0 and now f(x)=0 is not a function, it is solution of x.
In simple language, we can say by putting f(x) =0, we will get certain values of x, based on its degree.
For example let us say f(x) = x - 2
when we put f(x) =0, we will get x-2=0 --> x=2.

Thus, first statement which says "this will be the equation of x axis" is WRONG.

Second sentence "Integral of f(x) will be the area under the curve f(x) and the x axis." is absolutely correct.
The next sentence is "Since f(x) and x axis coincide , area under them will be 0." In this statement, f(x) and x axis are not coinciding, instead f(x)=0 gives solution of 'x'. And we cannot define area under the points.
"But antiderivative of 0 is a constant." - this sentence is correct antiderivative of 0 is constant and that constant can also be equal to 0.