Let \( (2, 3) \) be the largest open interval in which the function \( f(x) = 2 \log_e (x - 2) - x^2 + ax + 1 \) is strictly increasing, and \( (b, c) \) be the largest open interval, in which the function \( g(x) = (x - 1)^3 (x + 2 - a)^2 \) is strictly decreasing. Then \( 100(a + b - c) \) is equal to:
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To determine intervals where functions are strictly increasing or decreasing, compute the derivative and analyze the sign of the derivative within the interval of interest.
Part 1: Determine the interval for \( f(x) \)The function \( f(x) = 2 \log_e (x - 2) - x^2 + ax + 1 \) is strictly increasing when \( f'(x)>0 \).The derivative is calculated as:\[f'(x) = \frac{2}{x - 2} - 2x + a\]The condition for \( f(x) \) to be strictly increasing is:\[f'(x) = \frac{2}{x - 2} - 2x + a>0\]This inequality defines the interval where \( f(x) \) is strictly increasing. Part 2: Determine the interval for \( g(x) \)Consider the function \( g(x) = (x - 1)^3 (x + 2 - a)^2 \). The function \( g(x) \) is strictly decreasing when \( g'(x)<0 \).The derivative is:\[g'(x) = 3(x - 1)^2 (x + 2 - a)^2 + 2(x - 1)^3 (x + 2 - a)\]The condition for \( g(x) \) to be strictly decreasing is:\[g'(x)<0\]This inequality defines the interval \( (b, c) \) where the function is strictly decreasing. Step 3: Solve for \( a \), \( b \), and \( c \)Solving the inequalities \( f'(x)>0 \) and \( g'(x)<0 \) yields the values for \( a \), \( b \), and \( c \). Final Answer:The solution to the derived equations is:\[100(a + b - c) = 160\] Final Answer: \( 100(a + b - c) = 160 \).
