If the function  $f\left(x\right)=(4-p)x^3+(p-2)x^2+(p^2-25)x+2$ has a local minimum at some $x\in (-\infty ,0)$  and a local maximum at some  $x\in (0,\infty )$ then the true set of values of p is____

$$\begin{gathered} f\left(x\right)=(4-p)x^3+(p-2)x^2+(p^2-25)x+2 \\ {f}^1\left(x\right)=3(4-p)x^2+2(p-2)x+(p^2-25) \end{gathered}$$

One root positive and one root negative. 
Let $4 – p > 0;$

Here Minima occurs at $\alpha$ and maxima occurs at  $\beta$ which is allowed 

$$\begin{gathered} Let 4 – p < 0 \\ 4 < p \\ f\left(0\right) > 0; \\ {p}^2-25>0 \\ \Rightarrow p\in (-\infty ,-5)\cup (5,\infty ) \\ Thus, p\in (5,\infty ) \end{gathered}$$