Consider the polynomial f(x) = x 3 −3x 2 +x−3. One root of this polynomial is r = 3. Now consider the polynomial fϵ(x) = (1 + ϵ)x 3 − 3x 2 + x − 3, where ϵ is small.(a) Using the sensitivity formula for rootfinding, predict the shift in the root ∆r from r = 3 as a function of ϵ. What does this tell you about how errors in the coefficient of the cubic term of the input (backward error) are magnified in the output of rootfinding (forward error)

step-by-step explanation:

the system of inequalities is

step-by-step explanation:

step 1

find the equation of the dashed line

we have the points

(0,0) and (3,1)

the line passes through the origin, so, is a direct variation

find the slope

the equation of the dashed line is

the solution of the inequality a s the shaded area above the dashed line

therefore

the equation of the inequality a is

step 2

find the equation of the solid line

we have the points

(4,0) and (0,4)

find the slope

the equation of the solid line is

the solution of the inequality b s the shaded area below the solid line

therefore

the equation of the inequality b is

step 3

the system of inequalities is

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step-by-step explanation: