Mixing
Are following statements correct? [Extrema, derivatives]
$begingroup$
I have a few questions on the topic of extrema and derivatives:
- Consider a differentiable function $f:mathbb{R}tomathbb{R}$ that reaches its maximum in $ainmathbb{R}$. This implies that $f'(a)=0$.
I think this is true. We now that $f$ is differentiable over $mathbb{R}$, so $f'(a)$ exists. Because a (global) maximum is always a local maximum, we can conclude that $f'(a)=0$.
- If $f in C^1([0,1])$ reaches its maximum in a point $ain [0,1]$, then $f'(a)=0$.
This isn't necessarily true when $a=0$ or $a=1$, because then we're not working with an interior point ($f$ is differentiable over $]0,1[$).
Are my answers and way of thinking correct? Thanks for helping!
real-analysis
asked Jan 11 at 15:34
ZacharyZachary
1559
$endgroup$
add a comment |
$begingroup$
I have a few questions on the topic of extrema and derivatives:
- Consider a differentiable function $f:mathbb{R}tomathbb{R}$ that reaches its maximum in $ainmathbb{R}$. This implies that $f'(a)=0$.
I think this is true. We now that $f$ is differentiable over $mathbb{R}$, so $f'(a)$ exists. Because a (global) maximum is always a local maximum, we can conclude that $f'(a)=0$.
- If $f in C^1([0,1])$ reaches its maximum in a point $ain [0,1]$, then $f'(a)=0$.
This isn't necessarily true when $a=0$ or $a=1$, because then we're not working with an interior point ($f$ is differentiable over $]0,1[$).
Are my answers and way of thinking correct? Thanks for helping!
real-analysis
asked Jan 11 at 15:34
ZacharyZachary
1559
$endgroup$
add a comment |
$begingroup$
I have a few questions on the topic of extrema and derivatives:
- Consider a differentiable function $f:mathbb{R}tomathbb{R}$ that reaches its maximum in $ainmathbb{R}$. This implies that $f'(a)=0$.
I think this is true. We now that $f$ is differentiable over $mathbb{R}$, so $f'(a)$ exists. Because a (global) maximum is always a local maximum, we can conclude that $f'(a)=0$.
- If $f in C^1([0,1])$ reaches its maximum in a point $ain [0,1]$, then $f'(a)=0$.
This isn't necessarily true when $a=0$ or $a=1$, because then we're not working with an interior point ($f$ is differentiable over $]0,1[$).
Are my answers and way of thinking correct? Thanks for helping!
real-analysis
asked Jan 11 at 15:34
ZacharyZachary
1559
$endgroup$
I have a few questions on the topic of extrema and derivatives:
- Consider a differentiable function $f:mathbb{R}tomathbb{R}$ that reaches its maximum in $ainmathbb{R}$. This implies that $f'(a)=0$.
I think this is true. We now that $f$ is differentiable over $mathbb{R}$, so $f'(a)$ exists. Because a (global) maximum is always a local maximum, we can conclude that $f'(a)=0$.
- If $f in C^1([0,1])$ reaches its maximum in a point $ain [0,1]$, then $f'(a)=0$.
This isn't necessarily true when $a=0$ or $a=1$, because then we're not working with an interior point ($f$ is differentiable over $]0,1[$).
Are my answers and way of thinking correct? Thanks for helping!
real-analysis
real-analysis
asked Jan 11 at 15:34
ZacharyZachary
1559
asked Jan 11 at 15:34
ZacharyZachary
1559
asked Jan 11 at 15:34
ZacharyZachary
1559
asked Jan 11 at 15:34
ZacharyZachary
1559
asked Jan 11 at 15:34
ZacharyZachary
1559
1559
add a comment |
add a comment |
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