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Jun
5
comment Inequality with exponents
The differentiation step does not look right to me, we have either $ (a^x)' = \ln (a) \times a^x $ or $(x^a)'=a \times x^{a-1}$.
Jun
4
answered For what values of $a$ the equation $(ax)^2-x^4=e^{|x|}$ have no solution
Jun
4
comment Inequality with exponents
This happens for all positive real number and induction clearly does not work nicely here.
Jun
4
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?
Jun
4
awarded  Commentator
Jun
4
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?
Jun
4
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.
Jun
4
asked Inequality with exponents
Dec
6
awarded  Teacher
Jul
21
answered Percentages - Find Maximum value.
May
20
comment Differentiablity and Taylor polynomials
@Thomas So, what do I need to prove then. Cheers
May
20
asked Differentiablity and Taylor polynomials
May
5
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.
May
5
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 $f(x) \in B_{\epsilon}(f(x_0))$
May
5
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.
May
5
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.
May
5
awarded  Student
May
5
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 $.
May
5
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.
May
5
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.