$f(x)=0\mathrm{.}25x^2-2\mathrm{.}75x+6\mathrm{.}66$ fonksiyonu i$$in a$$a$$daki ad$$mlar$$ izleyin$$z ve soru$$a$$ cevaplaymiz.

a$)$ Python'da NumPy k$$t$$phanesi i$$erisindeki linspace komutunu kullanarak $0,10$ aral$$nd $1000$ noktada $x^{\text{'}}$i elde ediniz. Elde etti$$iniz $x$ ile $f(x)$ fonksiyonunun degerlerini elde ediniz ve Matplotlib k$$t$$phanesi i$$erisinde yer alan pyplot alt mod$$l$$n$$ kullanarak $f(x)$ fonksiyonunu $x\text{'}$e g$$re $$izdiriniz$.$izdirdi$$in$$z grafige izgaralar$($grid$)$ ekleyiniz ve fonksiyonunun k$$klerinin yerini ve bulunduklar aral$$ grafige bakarak tahmin ve tespit edip yaz$$n$$z$.$

b$)f(x)$ fonksiyonunun k$$klerini analitik olarak bulunuz ve $($a$)$ikk$$nda elde etti$$iniz grafi$$e bakarak tahmin etti$$iniz k$$klerle kar$$la$$t$$r$$nz$.$

c$)($a$)$kk$$ndaki bulgular$$n$$zdan hareketle derste i$$lenen t$$m k$$k bulma y$$ntemlerini sirasiyla $($Bisection$,$ Fixed Point Iteration, Newton ve Secant$)$ her bir k$$k$$n bulundu$$u aral$$k i$$in verilen $f(x)$ fonksiyonu i$$in ger$$ekleyiniz$.$ Bunun i$$in Python'da yazd$$n$$z her bir kod sat$$r$$nda yapt$$n$$z i$$lemi a$$klay$$n$$z$.$ Raporunuzda her bir y$$ntem i$$in iterasyon ad$$mlar$$n$$ g$$steren bir tablo olu$$turunuz ve k$$k$$ hangi hata tolerans$$yla ve ka$$nc$$ iterasyonda buldu$$unuzu belirtiniz.

d$)($b$)$kk$$nda farkl$$ k$$k bulma y$$ntemlerinden elde etti$$iniz t$$m k$$kleri listeleyiniz ve analitik hesaplad$$n$$z k$$k de$$erleriyle ba$$ hatalar$$m hesaplay$$n$$z$.$ Derste i$$lediklerinizi de g$$z $$n$$nde bulundurarak y$$ntemlerin hatalar$$n irdeleyiniz.

e$)$ Yukar$$daki t$$m $$klarda yapt$$klar$$n$$z$$ g$$zden ge$$iriniz$.$ Bulgular$$n$$z$$ ve $$rendiklerinizi yorumlay$$n$$z ve yaz$$n$$z$.$

For the function $f(x)=0\mathrm{.}25x^2-2\mathrm{.}75x+6\mathrm{.}66,$ follow the steps below and answer the questions:

a$)$ Using the linspace command in the NumPy library in Python, obtain $x$ in $1000$ points in the range $0,10.$ Obtain the values of $f(x)$ with the obtained $x,$ and plot $f(x)$ against $x$ using the pyplot submodule within the Matplotlib library. Add grids to the graph and make an estimate of the locations of the roots of the function by examining the graph, then write down your predictions.

b$)$ Find the roots of the $f(x)$ function analytically and compare them with the roots you estimated in part $($a$).$

c$)$ Based on the findings from part $($a$),$ implement all root$-$finding methods covered in the class $($Bisection$,$ Fixed Point Iteration, Newton, and Secant$)$ for the given $f(x)$ function in each root's respective interval. Explain each step in the Python code you write. Create a table in your report for each method showing the iteration steps and specify the error tolerance and the iteration number at which the root was found.

d$)$ List all the roots obtained from different root$-$finding methods in part $($b$)$ and calculate their relative errors compared to the analytically calculated root values. Examine the errors of the methods in consideration of what was covered in the class.

e$)$ Review all the steps you took in the above sections. Interpret and write down your findings and what you have learned.