Extreme value outside domain
$begingroup$
Hi I'm practicing finding extremas/monotonicity.
What if the derivative changes sign (function changes the monotonicity) at some point $x$ that is outside the domain?
Like, for example, $f(x) = x^3e^{frac{-1}{x}}$ the domain of derivative is $D_{f'} = (infty;0),(0,infty)$ and one of derivative's solutions is $x = 0$ which does not belong to neither domain of function $D_{f}$ nor domain of derivative $D_{f'}$. The function changes monotonicity, but does it have extreme value in there?
And similiar situation: What if the $x$ does belog to domain of $f$ but does not belong to domain of $f'$?
real-analysis calculus derivatives
asked Jan 2 at 14:31
wenoweno
36011
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add a comment |
$begingroup$
Hi I'm practicing finding extremas/monotonicity.
What if the derivative changes sign (function changes the monotonicity) at some point $x$ that is outside the domain?
Like, for example, $f(x) = x^3e^{frac{-1}{x}}$ the domain of derivative is $D_{f'} = (infty;0),(0,infty)$ and one of derivative's solutions is $x = 0$ which does not belong to neither domain of function $D_{f}$ nor domain of derivative $D_{f'}$. The function changes monotonicity, but does it have extreme value in there?
And similiar situation: What if the $x$ does belog to domain of $f$ but does not belong to domain of $f'$?
real-analysis calculus derivatives
asked Jan 2 at 14:31
wenoweno
36011
$endgroup$
add a comment |
$begingroup$
Hi I'm practicing finding extremas/monotonicity.
What if the derivative changes sign (function changes the monotonicity) at some point $x$ that is outside the domain?
Like, for example, $f(x) = x^3e^{frac{-1}{x}}$ the domain of derivative is $D_{f'} = (infty;0),(0,infty)$ and one of derivative's solutions is $x = 0$ which does not belong to neither domain of function $D_{f}$ nor domain of derivative $D_{f'}$. The function changes monotonicity, but does it have extreme value in there?
And similiar situation: What if the $x$ does belog to domain of $f$ but does not belong to domain of $f'$?
real-analysis calculus derivatives
asked Jan 2 at 14:31
wenoweno
36011
$endgroup$
Hi I'm practicing finding extremas/monotonicity.
What if the derivative changes sign (function changes the monotonicity) at some point $x$ that is outside the domain?
Like, for example, $f(x) = x^3e^{frac{-1}{x}}$ the domain of derivative is $D_{f'} = (infty;0),(0,infty)$ and one of derivative's solutions is $x = 0$ which does not belong to neither domain of function $D_{f}$ nor domain of derivative $D_{f'}$. The function changes monotonicity, but does it have extreme value in there?
And similiar situation: What if the $x$ does belog to domain of $f$ but does not belong to domain of $f'$?
real-analysis calculus derivatives
real-analysis calculus derivatives
asked Jan 2 at 14:31
wenoweno
36011
asked Jan 2 at 14:31
wenoweno
36011
asked Jan 2 at 14:31
wenoweno
36011
asked Jan 2 at 14:31
wenoweno
36011
asked Jan 2 at 14:31
wenoweno
36011
36011
add a comment |
add a comment |
1 Answer
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$begingroup$
What if the derivative changes sign (function changes the monotonicity) at some point $x$ that is outside the domain?
It doesn't: if a function doesn't exist at a certain point, it doesn't have a derivative there either.
Like, for example, $f(x) = x^3e^{frac{-1}{x}}$ the domain of derivative is $D_{f'} = (infty;0),(0,infty)$ and one of derivative's solutions is $x = 0$ which does not belong to neither domain of function $D_{f}$ nor domain of derivative $D_{f'}$.
How can $x=0$ be a zero of the derivative, if the derivative doesn't even exist in $x=0$...?
And similiar situation: What if the $x$ does belog to domain of $f$ but does not belong to domain of $f'$?
That's possible since a function isn't necessarily differentiable at every point of its domain. That doesn't mean it can't have an extreme value at such a point, think of $|x|$ at $x=0$ for example: no derivative, but the function has a minimum there.
answered Jan 2 at 14:38
StackTDStackTD
24.3k2254
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1 Answer
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1 Answer
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$begingroup$
What if the derivative changes sign (function changes the monotonicity) at some point $x$ that is outside the domain?
It doesn't: if a function doesn't exist at a certain point, it doesn't have a derivative there either.
Like, for example, $f(x) = x^3e^{frac{-1}{x}}$ the domain of derivative is $D_{f'} = (infty;0),(0,infty)$ and one of derivative's solutions is $x = 0$ which does not belong to neither domain of function $D_{f}$ nor domain of derivative $D_{f'}$.
How can $x=0$ be a zero of the derivative, if the derivative doesn't even exist in $x=0$...?
And similiar situation: What if the $x$ does belog to domain of $f$ but does not belong to domain of $f'$?
That's possible since a function isn't necessarily differentiable at every point of its domain. That doesn't mean it can't have an extreme value at such a point, think of $|x|$ at $x=0$ for example: no derivative, but the function has a minimum there.
answered Jan 2 at 14:38
StackTDStackTD
24.3k2254
$endgroup$
add a comment |
$begingroup$
What if the derivative changes sign (function changes the monotonicity) at some point $x$ that is outside the domain?
It doesn't: if a function doesn't exist at a certain point, it doesn't have a derivative there either.
Like, for example, $f(x) = x^3e^{frac{-1}{x}}$ the domain of derivative is $D_{f'} = (infty;0),(0,infty)$ and one of derivative's solutions is $x = 0$ which does not belong to neither domain of function $D_{f}$ nor domain of derivative $D_{f'}$.
How can $x=0$ be a zero of the derivative, if the derivative doesn't even exist in $x=0$...?
And similiar situation: What if the $x$ does belog to domain of $f$ but does not belong to domain of $f'$?
That's possible since a function isn't necessarily differentiable at every point of its domain. That doesn't mean it can't have an extreme value at such a point, think of $|x|$ at $x=0$ for example: no derivative, but the function has a minimum there.
answered Jan 2 at 14:38
StackTDStackTD
24.3k2254
$endgroup$
add a comment |
$begingroup$
What if the derivative changes sign (function changes the monotonicity) at some point $x$ that is outside the domain?
It doesn't: if a function doesn't exist at a certain point, it doesn't have a derivative there either.
Like, for example, $f(x) = x^3e^{frac{-1}{x}}$ the domain of derivative is $D_{f'} = (infty;0),(0,infty)$ and one of derivative's solutions is $x = 0$ which does not belong to neither domain of function $D_{f}$ nor domain of derivative $D_{f'}$.
How can $x=0$ be a zero of the derivative, if the derivative doesn't even exist in $x=0$...?
And similiar situation: What if the $x$ does belog to domain of $f$ but does not belong to domain of $f'$?
That's possible since a function isn't necessarily differentiable at every point of its domain. That doesn't mean it can't have an extreme value at such a point, think of $|x|$ at $x=0$ for example: no derivative, but the function has a minimum there.
answered Jan 2 at 14:38
StackTDStackTD
24.3k2254
$endgroup$
What if the derivative changes sign (function changes the monotonicity) at some point $x$ that is outside the domain?
It doesn't: if a function doesn't exist at a certain point, it doesn't have a derivative there either.
Like, for example, $f(x) = x^3e^{frac{-1}{x}}$ the domain of derivative is $D_{f'} = (infty;0),(0,infty)$ and one of derivative's solutions is $x = 0$ which does not belong to neither domain of function $D_{f}$ nor domain of derivative $D_{f'}$.
How can $x=0$ be a zero of the derivative, if the derivative doesn't even exist in $x=0$...?
And similiar situation: What if the $x$ does belog to domain of $f$ but does not belong to domain of $f'$?
That's possible since a function isn't necessarily differentiable at every point of its domain. That doesn't mean it can't have an extreme value at such a point, think of $|x|$ at $x=0$ for example: no derivative, but the function has a minimum there.
answered Jan 2 at 14:38
StackTDStackTD
24.3k2254
answered Jan 2 at 14:38
StackTDStackTD
24.3k2254
answered Jan 2 at 14:38
StackTDStackTD
24.3k2254
answered Jan 2 at 14:38
StackTDStackTD
24.3k2254
24.3k2254
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