A question related to the mean value theorem
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Let $f$ be continuous on $[a,b]$ and differentiable on $(a,b)$. It it true that for every $cin[a,b]$, there exists some $(a_0, b_0)$ such that
$$f'(c) = frac{f(b_0)-f(a_0)}{b_0 - a_0},? $$
calculus derivatives continuity examples-counterexamples
edited Dec 20 '18 at 2:13
Batominovski
33.1k33293
asked Dec 20 '18 at 1:26
OneTwoOneOneTwoOne
307
$endgroup$
add a comment |
$begingroup$
Let $f$ be continuous on $[a,b]$ and differentiable on $(a,b)$. It it true that for every $cin[a,b]$, there exists some $(a_0, b_0)$ such that
$$f'(c) = frac{f(b_0)-f(a_0)}{b_0 - a_0},? $$
calculus derivatives continuity examples-counterexamples
edited Dec 20 '18 at 2:13
Batominovski
33.1k33293
asked Dec 20 '18 at 1:26
OneTwoOneOneTwoOne
307
$endgroup$
$begingroup$
You need to ensure that there are no points of inflection for $f$ in $(a, b) $ otherwise the result is false.
$endgroup$
– Paramanand Singh
Dec 20 '18 at 7:00
add a comment |
$begingroup$
Let $f$ be continuous on $[a,b]$ and differentiable on $(a,b)$. It it true that for every $cin[a,b]$, there exists some $(a_0, b_0)$ such that
$$f'(c) = frac{f(b_0)-f(a_0)}{b_0 - a_0},? $$
calculus derivatives continuity examples-counterexamples
edited Dec 20 '18 at 2:13
Batominovski
33.1k33293
asked Dec 20 '18 at 1:26
OneTwoOneOneTwoOne
307
$endgroup$
Let $f$ be continuous on $[a,b]$ and differentiable on $(a,b)$. It it true that for every $cin[a,b]$, there exists some $(a_0, b_0)$ such that
$$f'(c) = frac{f(b_0)-f(a_0)}{b_0 - a_0},? $$
calculus derivatives continuity examples-counterexamples
calculus derivatives continuity examples-counterexamples
edited Dec 20 '18 at 2:13
Batominovski
33.1k33293
asked Dec 20 '18 at 1:26
OneTwoOneOneTwoOne
307
edited Dec 20 '18 at 2:13
Batominovski
33.1k33293
asked Dec 20 '18 at 1:26
OneTwoOneOneTwoOne
307
edited Dec 20 '18 at 2:13
Batominovski
33.1k33293
edited Dec 20 '18 at 2:13
Batominovski
33.1k33293
edited Dec 20 '18 at 2:13
Batominovski
33.1k33293
33.1k33293
asked Dec 20 '18 at 1:26
OneTwoOneOneTwoOne
307
asked Dec 20 '18 at 1:26
OneTwoOneOneTwoOne
307
asked Dec 20 '18 at 1:26
OneTwoOneOneTwoOne
307
307
$begingroup$
You need to ensure that there are no points of inflection for $f$ in $(a, b) $ otherwise the result is false.
$endgroup$
– Paramanand Singh
Dec 20 '18 at 7:00
add a comment |
$begingroup$
You need to ensure that there are no points of inflection for $f$ in $(a, b) $ otherwise the result is false.
$endgroup$
– Paramanand Singh
Dec 20 '18 at 7:00
$begingroup$
You need to ensure that there are no points of inflection for $f$ in $(a, b) $ otherwise the result is false.
$endgroup$
– Paramanand Singh
Dec 20 '18 at 7:00
$begingroup$
You need to ensure that there are no points of inflection for $f$ in $(a, b) $ otherwise the result is false.
$endgroup$
– Paramanand Singh
Dec 20 '18 at 7:00
add a comment |
1 Answer
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No, not necessarily. Consider for example $f(x)=x^3$. Then $f'(0)=0$, but there are no $a$ and $b$ such that
$$frac{a^3-b^3}{a-b}=0$$
without having $a=b$.
answered Dec 20 '18 at 1:35
user628021user628021
962
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add a comment |
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1 Answer
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active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
No, not necessarily. Consider for example $f(x)=x^3$. Then $f'(0)=0$, but there are no $a$ and $b$ such that
$$frac{a^3-b^3}{a-b}=0$$
without having $a=b$.
answered Dec 20 '18 at 1:35
user628021user628021
962
$endgroup$
add a comment |
$begingroup$
No, not necessarily. Consider for example $f(x)=x^3$. Then $f'(0)=0$, but there are no $a$ and $b$ such that
$$frac{a^3-b^3}{a-b}=0$$
without having $a=b$.
answered Dec 20 '18 at 1:35
user628021user628021
962
$endgroup$
add a comment |
$begingroup$
No, not necessarily. Consider for example $f(x)=x^3$. Then $f'(0)=0$, but there are no $a$ and $b$ such that
$$frac{a^3-b^3}{a-b}=0$$
without having $a=b$.
answered Dec 20 '18 at 1:35
user628021user628021
962
$endgroup$
No, not necessarily. Consider for example $f(x)=x^3$. Then $f'(0)=0$, but there are no $a$ and $b$ such that
$$frac{a^3-b^3}{a-b}=0$$
without having $a=b$.
answered Dec 20 '18 at 1:35
user628021user628021
962
answered Dec 20 '18 at 1:35
user628021user628021
962
answered Dec 20 '18 at 1:35
user628021user628021
962
answered Dec 20 '18 at 1:35
user628021user628021
962
962
add a comment |
add a comment |
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$begingroup$
You need to ensure that there are no points of inflection for $f$ in $(a, b) $ otherwise the result is false.
$endgroup$
– Paramanand Singh
Dec 20 '18 at 7:00