Let $f:\mathbb{R}\to\mathbb{R}$ be a given function such that $f(0)=-1$. Prove or disprove: if $f'(x)>0$ and $f''(x)>0$ for every $x\in\mathbb{R}$, then there exist $x_0\in\mathbb{R}$ such that $f(x_0)=0$.
Calculus question about first and second derivatives
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calculus
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0For context: how likely do you think the proposition is to be true? – 2017-02-21
1 Answers
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Hint: You have strictly convex function. For any strictly convex function $f(x)>f(a)+f'(a)(x-a)$. Now take $a=0$ and $f'(a)>0$ and send $x\rightarrow\infty$ to show that the function diverges and since it is continuous, it must visit $0$.