Assume $$\{X_i\}\stackrel{\mathrm{i.i.d.}}{\sim} \mathcal{Uniform}(0,1)$$ Find the joint p.d.f. of $$X_{(n)} \hat= \max \{X_1,X_2,\ldots,X_5\}\quad\text{ and }\quad \bar X\hat=\sum^n_{i=1}{X_i}$$
The motivation of this question: If a line of length 1 is randomly divided into five parts, computing the probability of there exist one part with length more than 1/4, i.e. find $\Pr\{X_{(n)}\ge \frac{5\bar X}{4} \}$ with the setting above.
$X_1, \dotsc,X_n$ is iid from the uniform distribution on $[0,1]$. The maximum is $X_{(n)}$ which has a distribution with density $f(u)=n u^{n-1}$. Represent the joint density of the mean and the maximum as $f(m, u)=f(m \mid u) f(u)$ where $f(m \mid u)$ represents the conditional density of the mean given the maximum.
The order statistics $X_{(1)},\dotsc, X_{(n-1)}$, conditional on the maximum behaves as order statistics from an iid sample of size $n-1$ from the uniform distribution on $[0,u]$, so we can use that and the representation of the mean as $$ \bar{X}_n = \frac{u}{n}+ \frac{n-1}{n}\bar{X}_{n-1} $$ when the maximum $u$ is given. Then the distribution of $\bar{X}_{n-1}$ is a scaled version of the Irwin-Hall distribution, see also Can we make the Irwin-Hall distribution more general?. I will leave it at that, you should be able to complete.
