问题318: Consider the function f(x)=441 x^4-1596 x^3+1654 x^2-380 x+25.

Consider the function $f(x)=441 x^4-1596 x^3+1654 x^2-380 x+25$.
(i) Prove (using Ist derivative test) that there is a local maximum at $x \in(0.5,1.5)$.
(ii) Find the $x$-coordinate of the maximum point using Newton-method. Use initial guess $x_1=1.0$. Correct your answer to 4 decimal places.
(iii) Prove (using 2 nd derivative test) that this function has an inflection point at $x \in(1.0,1.5)$.