Question

For the polynomial function ƒ(x) = x4 − 12x3 + 46x2 − 60x + 25, find all local and global extrema.

Answer 1

$\bf f(x)=x^4-12x^3+46x^2-60x+25\\\\ -------------------------------\\\\ \cfrac{df}{dx}=4x^3-36x^2+92x-60\implies 0=4x^3-36x^2+92x-60 \\\\\\ 0=x^3-9x^2+23x-15\impliedby \textit{doing some synthetic division with 5} \\\\\\ 0=(x^2-4x+3)(x-5)\implies 0=(x-3)(x-1)(x-5)\\\\ -------------------------------\\\\ \begin{cases} f(1)=0\leftarrow minimum&1,0\\ f(3)=16\leftarrow maximum&3,16\\ f(5)=0\leftarrow minimum&5,0 \end{cases}\impliedby \textit{3 extrema, locals and globals}$