GATE | GATE-CS-2003 | Question 42

A piecewise linear function f(x) is plotted using thick solid lines in the figure below (the plot is drawn to scale).

If we use the Newton-Raphson method to find the roots of f(x) = 0 using x0, x1 and x2 respectively as initial guesses, the roots obtained would be
(A) 1.3, 0.6, and 0.6 respectively
(B) 0.6, 0.6, and 1.3 respectively
(C) 1.3, 1.3, and 0.6 respectively
(D) 1.3, 0.6, and 1.3 respectively


Answer: (D)

Explanation: First of all, There is a mistake in coordinates of a given point. I have corrected that in red color.

Now in Newton-Raphson method, we draw a tangent from our guess point, and our new guess would be the point where this tangent cuts x-axis. Now we choose initial guess points one by one :

x0 : Tangent at this point is line AB itself, and that would cut x-axis at 
     point (1.0,0.0)  (found using equation of line AB). So our next guess 
     would be 1.0. Point on the curve corresponding to this new guess 1.0 
    is shown as F. Now tangent at point F is line DE,   which cuts x-axis at
    1.3, and at this point, value of function is zero, so we found  the root 
    as 1.3.

x1 : Tangent at this point is line BE, which cuts x-axis at 0.6, also function 
     value is zero here, so we find root as 0.6.

x2 : Tangent at this point is line CD, which cuts x-axis at 1.05 (again found 
     by finding equation of line CD). Point on the curve corresponding to this
     new guess 1.05 is shown as G. Now tangent at point G is line DE, which cuts
     x-axis at 1.3, and at this point, value of function is zero, so we found 
    the root as 1.3. 

Source: Question 60 of http://www.cse.iitd.ac.in/~mittal/gate/gate_math_2003.html

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