Explanation? I know I need to take the derivative first just not sure how. Consider the function f(x) = xe^2x. Find a critical point. Does f have a relative maximum, a relative minimum, or neither at that point?

Question

emmatassone · Answer

First take the derivative:$$f(x)=x.e ^{2x}\rightarrow f \prime(x)= e ^{2x}+x(2e ^{2x})= e ^{2x}.(1+2x)$$ now we equal to zero the equation to find critical points:$$f \prime(x)=0 \rightarrow e ^{2x}(1+2x)=0\rightarrow 1+2x=0\rightarrow x=-\frac{ 1 }{ 2 }$$ we used $$e ^{2x}\neq0  (\forall x \in \mathbb{R})$$ finally  evaluate x=-1/2 in the second derivative to see if its a maximum or minimum. Here its a graph to corroborate the solution.

anonymous · Answer

Can you explain how you got the derivative? Is there a formula or something to use?

emmatassone · Answer

there is a formula to derivative a multiplication:$$f(x)= g(x).h(x)\rightarrow f \prime(x) = g \prime(x).h(x) + g(x).h \prime(x)$$ and there is also a formula to get the derivative  of each funtion. you can look for them in internet

anonymous · Answer

Thanks!