I have $$ f(x)=x^7-12 $$ and need to find the roots with $\epsilon=0.0001$ accuracy. I have done the first steps but can't continue further. Here's what I've done:
Ive narrowed down the range where I there is a root and found those $x$ values: $$f(1.4)=-1.45, f(1.43)=0.227$$ So I have $$\Omega a=[1.4;1.43]$$ $$f \prime(x) = 7x^6, f \prime \prime(x) = 42x^5$$ $${\frac{1}{|f \prime(x)|}} = max{\frac{1}{|7x^6|}} = (x=1.4) = {\frac{1}{52.7}} = a_1,$$ $${\frac{|f \prime \prime(x)|}{1}} = max|21x^5| = (x=1.43)= 125.5 = a_2$$ $$c = a_1a_2 = {\frac{125.5}{52.7}} = 2.38$$ Choose $$a = {\frac{1.4+1.43}{2}}=1.41$$ $$b=min(a, {\frac{1}{c}}) = min(1.41, {\frac{1}{2.38}})$$ The minimum is the value of $c$, and our teacher said that we need to apply such steps after which the minimum is the value of $a$. What am I doing wrong?
Okay, I got what's wrong. I chose $a$ incorrectly. Should be $$ a={\frac{1.43-1.4}{2}} = {\frac{0.03}{2}} = 0.015 $$ In that case we can have $b$ to be the value of $a$ $$b=min(a, {\frac{1}{c}}) = min(0.015, {\frac{1}{2.38}}) = 0.015$$