Original image: https://i.stack.imgur.com/2d7tt.png
$30$. For what positive value of $c$ does the equation $\log x = cx^4$ have exactly one real solution for $x$?
$\quad \text{(A) } \frac{1}{4e} \qquad \text{(B) } \frac{1}{4e^4} \qquad \text{(C) } \frac{e^4}{4} \qquad \text{(D) } \frac{4}{e^{1/4}} \qquad \text{(E) } 4e^{1/4} $
I'm stuck in this problem simply because I never encounter this kind before. So do we have a general approach to question like this?
I do not need an answer, I just want to know how to approach this kind of problem.
THank you for your help.
First step is to draw a sketch of the graphs of $\log x$ and $cx^4$. From here it is clear what the question is really asking: We need to have the two graphs intersect, and we also need the intersection to be tangent (otherwise, the two graphs will miss each other, or they will meet twice).
The rest is calculus (really, mostly algebra), $$ \begin{cases}\log x=cx^4\\ \frac1x=4cx^3 \end{cases}\\ \implies c=\frac{1}{4x^4}\implies \log x=1/4\implies x=e^{1/4} $$ so $c=\frac{1}{4e}$.