Not homework, just curious

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Main Question or Discussion Point

*Let [itex] f\left( x \right) [/itex] be a twice-differentiable function for which
[tex] \; \mathop {\lim }\limits_{x \to 0^ - } f\left( x \right) = \infty [/tex]

Then, is it true that
[tex] \mathop {\lim }\limits_{x \to 0^ - } f\,{''}\left( x \right) > 0\;? [/tex]

----------------------------------------------
Or a little differently,

*Let [itex] f\left( x \right) [/itex] be a twice-differentiable function for which
[tex]\mathop {\lim }\limits_{x \to 0^ - } f\left( x \right) = \infty \;{\text{and }}\forall x < 0,f\,'\left( x \right) > 0 [/tex]

Then, is it true that
[tex] \mathop {\lim }\limits_{x \to 0^ - } f\,{''}\left( x \right) > 0\;? [/tex]

Just curious|:redface:
 
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Answers and Replies

  • #2
VietDao29
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You can prove it by contradiction.
So let f(x) be a function that is continuous, increasing in [-δ, 0), its first and second derivatives are also continuous in [-δ, 0), where δ > 0, so f(-δ) <> +∞, f'(-δ) <> +/-∞, f''(-δ) <> +/-∞.
So assume [tex]\lim_{x \rightarrow 0 ^ -} f''(x) <= 0[/tex]
Now [tex]\lim_{x \rightarrow 0 ^ -} f''(x) = 0[/tex] means that there exists ε > 0, and 0 <= σ < ∞, such that, [itex]\forall x \in (-\varepsilon,\ 0), \ f'(x) \leq \sigma[/itex], where 0 <= σ < ∞
Since f(-ε) <> +/-∞, so f(0) <= f(-ε) + σε < + ∞.
Now [tex]\lim_{x \rightarrow 0 ^ -} f''(x) < 0[/tex] means that there exists ε > 0 such that, [itex]\forall x_1, x_2 \in (-\varepsilon,\ 0) \mbox{ and } \ x_1 < x_2 \rightarrow \ f'(x_1) > f'(x_2)[/itex]
Let [tex]x_2 = \lim_{\beta \rightarrow 0 ^ -} 0 + \beta[/tex]
Let α be the slope of the line that connects two points (x1, f(x1)), and (x2, f(x2)), so
f'(-ε) > f'(x1) > α > f'(x2) > 0 (f(x) is increasing), so α <> +/-∞.
(You can use: [tex]\forall \alpha , \ \beta | \alpha < \beta, \exists \gamma | \ \alpha < \gamma < \beta,\frac{f(\alpha) - f(\beta)}{\alpha - \beta} = f'(\gamma)[/tex] to prove that).
So f(x2) = f'(x1) + α(x2 - x1) <> + ∞, f(x2) does not tend to infinity.
(Q.E.D).
So, if:
[tex]\lim_{x \rightarrow 0 ^ -} f(x) = + \infty[/tex], then:
[tex]\lim_{x \rightarrow 0 ^ -} f''(x) > 0[/tex].
Note that it should be + ∞, (not - ∞).
Do the same, and you will get:
[tex]\lim_{x \rightarrow 0 ^ -} f(x) = - \infty[/tex], then:
[tex]\lim_{x \rightarrow 0 ^ -} f''(x) < 0[/tex].
To me the 0- is can be change to γ- for a more general case, where [itex]\gamma \neq \pm \infty[/itex]. It's still true.
Viet Dao,
 
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