The function f(x)=2 log(x-2)-x^2+4 x+1 increases on the interval (a) (1,2) (b) (2,3) (c) ((5)/(2

Step 1:u000AFirst, we need to find the derivative of the function $f(x)$. The derivative of a functi

The function f(x)=2 log(x-2)-x^2+4 x+1 increases on the...

Key Concepts

Domain of Logarithmic Functions

When a function includes a logarithmic term such as log(x - a), it is essential to recognize that the argument of the logarithm must be positive. This restriction (x - a > 0) defines the domain of the function and eliminates portions of the real line from consideration, as the logarithm is undefined for non-positive values.

Monotonicity and the First Derivative Test

Determining where a function is increasing involves analyzing its first derivative. The function is increasing on intervals where its derivative is positive. This concept, known as the First Derivative Test, enables one to identify the intervals of increase or decrease by solving the inequality f'(x) > 0.

Differentiation of Composite and Combined Functions

In functions that combine logarithmic and polynomial expressions, such as those involving log terms and quadratic terms, it is important to apply the rules of differentiation—such as the chain rule for composite functions and basic derivative rules for polynomials—to accurately determine the derivative. The resulting expression is then used to analyze the function’s behavior.

Solving Inequalities Involving Derivatives

After finding the derivative, solving inequalities is necessary to identify the specific intervals where the derivative is positive. This process typically involves combining and simplifying terms, handling rational expressions, and sometimes solving quadratic inequalities, which are key skills in analyzing the increasing or decreasing nature of a function.

Please give Ace some feedback