Viet Hoang Quoc
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 Jun5 comment Inequality with exponents The differentiation step does not look right to me, we have either $(a^x)' = \ln (a) \times a^x$ or $(x^a)'=a \times x^{a-1}$. Jun4 answered For what values of $a$ the equation $(ax)^2-x^4=e^{|x|}$ have no solution Jun4 comment Inequality with exponents This happens for all positive real number and induction clearly does not work nicely here. Jun4 comment Inequality with exponents Can you elaborate a bit more because I have used AM-GM but the difference in the power could be a major issue? Jun4 awarded Commentator Jun4 comment For what values of $a$ the equation $(ax)^2-x^4=e^{|x|}$ have no solution What do you mean by $|a| < \sqrt{2}$?. Do we have to prove that when $|a|<\sqrt{2}$, the equation has no solution? Jun4 comment For what values of $a$ the equation $(ax)^2-x^4=e^{|x|}$ have no solution The $\ln$ operation is not correct because it is applied for division or multiplication. Jun4 asked Inequality with exponents Dec6 awarded Teacher Jul21 answered Percentages - Find Maximum value. May20 comment Differentiablity and Taylor polynomials @Thomas So, what do I need to prove then. Cheers May20 asked Differentiablity and Taylor polynomials May5 comment Continuity leads to constant function (Assignment question) Here is another similar type of question proving the function is constant using the continuity definition, Let $f: [-1,1] \to \mathbb{R}$ be a function with $f(x)=f(x^2)$ for all $x \in (-1,1)$. Suppose that $f$ is continuous at 0. Prove that $f$ is a constant. May5 comment Continuity leads to constant function (Assignment question) Thanks all the the great effort made, let's me just recap what we have had so far: $f$ is continuous iff for all $\epsilon >0$, $x \in B_{\delta}(x_0)$ then $f(x) \in B_{\epsilon}(f(x_0))$ May5 comment Continuity leads to constant function (Assignment question) To my understanding, if we can pick a rational point $x_1$ satisfying $|x_1-x_0|< \delta$ such that $|f(x_1)-f(x_0)| \ge \epsilon$ then this will give a contradiction. May5 comment Continuity leads to constant function (Assignment question) @Benjamin Lim : This is the first course in Analysis so we have not learnt anything such as Topological spaces and other things, just simply the definition and way to go. Thanks. May5 awarded Student May5 comment Continuity leads to constant function (Assignment question) The formal one as follows For all $\epsilon >0$, there exists $\delta >0$ such that for all $x \in \mathbb{R}$ satisfying $|x-x_0| < \delta$, one has $|f(x)-f(x_0)| < \epsilon$. May5 comment Continuity leads to constant function (Assignment question) Yes, it was my initial idea of using Density Theorem. However, I do not see the connection between that with the function. May5 comment Continuity leads to constant function (Assignment question) Great solution, however, the question requires the answer to use the definition of the continuity. Thereby, I just do not know where to start.