Question

Show that the function (function below) is constant on the interval (0, +♾)

Answer 1

If the function is constant, that implies that the derivative is 0

`\begin{gathered} \Rightarrow(dF(x))/(dx)=(d)/(dx)\int_x^(4x)(1)/(t)dt \\ \\ \text{ using fundamental theorem of calculus} \\ \\ \Rightarrow(d)/(dx)(\int_0^(4x)(1)/(t)dt-\int_0^x(1)/(t)dt) \end{gathered}`
`\begin{gathered} \text{ using chain rule and fundamental theorem of calculus} \\ \\ \Rightarrow(d)/(dx)\int_0^(4x)(1)/(t)dt=(d)/(du)\int_0^u(1)/(t)dt\cdot(d(4x))/(dx)=4\cdot(1)/(u)=4\cdot(1)/(4x)=(1)/(x) \end{gathered}`

Also,

`(d)/(dx)\int_0^x(1)/(t)dt=(1)/(x)`

Then,

Since the derivative is 0, It is obvious that F(x) is a constant function.