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A Problem regarding the Mean Value Property of functions .

score 61 · Answer 1 · 2023-06-06 22:31:27

61

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A Problem regarding the Mean Value Property of functions .

Published on 06 Jun 2023 - 22:31

score 79 · Answer 2 · 2023-06-10 10:48:45

79

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Let $f:[0,1] \to \mathbb R$ differentiable s.t $f(0)=0$. $\forall x \in [0,1]$: $|f^{'}(x)| \le |f(x)|$. Prove $f(x) =0$, $\forall x \in [0,1]$.

Published on 10 Jun 2023 - 10:48

score 52 · Answer 3 · 2023-06-10 15:05:04

52

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Prove that $\sqrt{1+x} < 1+\frac{1}{2}x$ if $x>0$ using MVT

Published on 10 Jun 2023 - 15:05

score 173 · Answer 4 · 2023-06-15 16:41:47

173

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Prove that the function $g(x)=\begin{cases} \sin (\frac 1x)& x \neq 0 \\ 1& x=0 \end{cases}$ has no antiderivative

Published on 15 Jun 2023 - 16:41

score 40 · Answer 5 · 2023-06-20 12:04:22

40

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Differentiation,Mean Value Theorem,Real Analysis

Published on 20 Jun 2023 - 12:04

score 74 · Answer 6 · 2023-06-21 12:51:33

74

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Regarding averages and mean value theorem for integrals

Published on 21 Jun 2023 - 12:51

score 19 · Answer 7 · 2023-06-26 18:22:50

19

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Langrange Mean Value Theorem on tanx at x belongs to (0,π/2)

Published on 26 Jun 2023 - 18:22

score 65 · Answer 8 · 2023-06-29 17:16:36

65

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M.V.T. (Mean Value Theorem) application

Published on 29 Jun 2023 - 17:16

score 83 · Answer 9 · 2023-07-04 14:43:18

83

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Let $f, g$ be functions differentiable at $1$ and such that $f(x)+x ≤ g(x)+x^2$ and $f(1) = g(1).$ Show that $f'(1)+1 = g' (1)$

Published on 04 Jul 2023 - 14:43

score 52 · Answer 10 · 2023-07-07 12:20:19

52

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Prove that if $ f' $ is injective and $ L $ is the line tangent to $ x_0 $, then $ L $ and the graph of $ f $ don't intersect anywhere else

Published on 07 Jul 2023 - 12:20

score 40 · Answer 11 · 2023-07-10 10:43:25

40

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How to prove the existence of a mean value when it is difficult to constructing a primitive function?

Published on 10 Jul 2023 - 10:43

score 98 · Answer 12 · 2023-07-19 09:29:18

98

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Generalized Mean Value Theorem + Intermediate Value Theorem

Published on 19 Jul 2023 - 9:29

score 76 · Answer 13 · 2023-07-19 13:42:05

76

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Mean Value Theorem for semi-differentiable multivariate functions

Published on 19 Jul 2023 - 13:42

score 59 · Answer 14 · 2023-07-23 16:27:12

59

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If a partial derivative of a multivariable function exist does it implies that the function is continuous wrt the concerned variable ? Rudin $9.40$

Published on 23 Jul 2023 - 16:27

score 49 · Answer 15 · 2023-07-27 15:17:42

49

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Question about increasing, differentiable, strictly convex real function where either there is a (counter-)example, or Darboux Theorem probably helps

Published on 27 Jul 2023 - 15:17

List Question

A Problem regarding the Mean Value Property of functions .

Let $f:[0,1] \to \mathbb R$ differentiable s.t $f(0)=0$. $\forall x \in [0,1]$: $|f^{'}(x)| \le |f(x)|$. Prove $f(x) =0$, $\forall x \in [0,1]$.

Prove that $\sqrt{1+x} < 1+\frac{1}{2}x$ if $x>0$ using MVT

Prove that the function $g(x)=\begin{cases} \sin (\frac 1x)& x \neq 0 \\ 1& x=0 \end{cases}$ has no antiderivative

Differentiation,Mean Value Theorem,Real Analysis

Regarding averages and mean value theorem for integrals

Langrange Mean Value Theorem on tanx at x belongs to (0,π/2)

M.V.T. (Mean Value Theorem) application

Let $f, g$ be functions differentiable at $1$ and such that $f(x)+x ≤ g(x)+x^2$ and $f(1) = g(1).$ Show that $f'(1)+1 = g' (1)$

Prove that if $ f' $ is injective and $ L $ is the line tangent to $ x_0 $, then $ L $ and the graph of $ f $ don't intersect anywhere else

How to prove the existence of a mean value when it is difficult to constructing a primitive function?

Generalized Mean Value Theorem + Intermediate Value Theorem

Mean Value Theorem for semi-differentiable multivariate functions

If a partial derivative of a multivariable function exist does it implies that the function is continuous wrt the concerned variable ? Rudin $9.40$

Question about increasing, differentiable, strictly convex real function where either there is a (counter-)example, or Darboux Theorem probably helps

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