How should I show that there exists only one solution?

[Math100]

Homework Statement

Consider the equation $g(a)=\Gamma$ where $g(a)=(c-1)a+\frac{3}{4}a^3$.
Show that for $c\geq 1$, only one solution exists for $\Gamma>0$.

Relevant Equations

None.

So far, I've got that $g(a)=(c-1)a+\frac{3}{4}a^3\implies g'(a)=c-1+\frac{9}{4}a^2$.
I know that if the first derivative of a function is positive (greater than $0$),
then that function is always/strictly increasing. However, how should I construct
this proof in order to show that there exists only one solution with the given quantifiers of
$\Gamma>0, c\geq 1$? What needs to be done?

 

[person_random_normal]

When a tends to negative infinity, the function tends to negative infinity, and when a tends to positive infinity, the function tends to positive infinity. Therefore, since the function's first derivative is always positive, the given equation will have only one solution.

 

[Math100]

Thank you for the help!

 

[vela]

You need to be a bit more careful. If $c=1$, which is allowed, then $g'(0) = 0$, so the function isn't always strictly increasing. You should deal with this case as well in your proof.