True or false:
let f(x) be a non-negative continuous function in [1, $\infty$] and also we know that $\int_1^\infty f(x)dx < \infty $
Assuming f has a continuous derivative, and $\int_1^\infty f' < \infty$ does it mean $\lim_{x\to \infty} f(x)=0?$
#Calculus
runic sky
worn finchBOT
True or false:
let f(x) be a non-negative continuous function in [1, $\infty$] ...
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mtr123
nova vale
any hint about the proof?
we have a function that's always non negative (never goes below zero)
the area under this function from 1 to infinity is finite
the function has a smooth derivative
the area under the derivative is also finite
can you imagine what would happen if the function diddnt go to zero ?
runic sky
can you imagine what would happen if the function diddnt go to zero ?
if f doesn't go to 0 we have that int f' between a to b is gonna be f(b)-f(a) which is gonna be > 0 so i think the int of f' will diverge?
nova vale
if f doesn't go to 0 we have that int f' between a to b is gonna be f(b)-f(a) wh...
exactly
But this contradicts our given condition tTherefore f(x) must go to 0
runic sky
so just take that and take b to infty? look for the limit?
quiet pilotBOT
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nova vale
youre welcome