Both [tex]f[/tex] and [tex]g[/tex] satisfy the conditions for Rolle's theorem, which then says there exists [tex]c\in[-1,3][/tex] (for [tex]f[/tex]) such that [tex]f'(c)=0[/tex], and [tex]c\in[-2,1[/tex] (for [tex]g[/tex]) such that [tex]g'(c)=0[/tex].
1.
[tex]f(x)=x^2-2x-8\implies f'(x)=2x-2[/tex]
[tex]f'(c)=2c-2=0\implies c=1[/tex]
2.
[tex]g(t)=2t-t^2-t^3=0\implies g'(t)=2-2t-3t^2[/tex]
[tex]g'(c)=2-2c-3c^2=0\implies c=\dfrac{-1\pm\sqrt7}3[/tex]
