# Define domain and intervals of rising/falling

• Jan 23rd 2009, 12:45 AM
courteous
Define domain and intervals of rising/falling
"Define domain and intervals of rising/falling of the following function f(x): $f(x)=\int_{0}^{x}\log(t^3+1)dt$."

Domain of $f(x)$ is where $t>-1$: $(-1, \infty)$

What about function falling/rising (assuming that I got the domain right(Happy))? Isn't it so, that you just take derivative of f(x) and that "cancels" the integral, which would mean that f(x) is always rising for $t\geq 0$, because its derivative ( $\log(t^3+1)$) is always positive with $t\geq 0$?
• Jan 23rd 2009, 01:46 AM
mr fantastic
Quote:

Originally Posted by courteous
"Define domain and intervals of rising/falling of the following function f(x): $f(x)=\int_{0}^{x}\log(t^3+1)dt$."

Domain of $f(x)$ is where $t>-1$: $(-1, \infty)$

What about function falling/rising (assuming that I got the domain right(Happy))? Isn't it so, that you just take derivative of f(x) and that "cancels" the integral, which would mean that f(x) is always rising for $t\geq 0$, because its derivative ( $\log(t^3+1)$) is always positive with $t\geq 0$?

f(x) is 'rising' for values of x such that f'(x) > 0 and 'falling' for values of x such that f'(x) < 0.

Note that $f'(x) = \log (x^3 + 1)$. And note that $\log (x^3 + 1) > 0$ for $x > 0 \, ....$