Final answer:
To find the x-value for the maximum of the quadratic function, it is typically found at x=-b/(2a). However, using Newton's method with an initial guess x0=1 and applying one iteration, the second guess for the root is found to be x1=1/2, making it the correct answer (a) 1/2.
Step-by-step explanation:
The student has been asked to find the x-value where the maximum value of f(x)=2x^2-4x+3 occurs, without using a calculator. To solve for the maximum or minimum of a quadratic equation, one can use the vertex formula, which states that the x-coordinate of the vertex (and thus the maximum or minimum) is at x=-b/(2a) for a quadratic equation ax^2 + bx + c. In this case, a=2, b=-4, and the x-value of the vertex is x=1. However, it's also noted to use Newton's method with an initial guess of x0=1. Applying Newton's method once would give us the second guess (which is the answer the question is looking for).
Newton's method uses iterations to find successively better approximations to the roots (or zeroes) of a real-valued function. The formula for Newton's method is xn+1 = xn - f(xn)/f'(xn). For the given function f(x), the derivative f'(x) = 4x-4. Plugging in the first guess x0=1, we get the second guess as x1 = 1 - (2(1)^2 - 4(1) + 3)/(4(1) - 4) = 1/2, making our answer (a) 1/2.