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

step 1 in this problem we have f x is equal to 2 log of x minus 2 minus x square plus 4x plus 1 now not

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

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Key Concepts

Inequalities and Critical Points

After finding the derivative, solving inequalities (setting the derivative greater than zero) helps identify the intervals over which the function is increasing. Critical points, where the derivative changes sign, partition the domain and are essential in determining the monotonic behaviour of the function.

Logarithmic Differentiation

Differentiating a logarithmic function involves applying the chain rule. When differentiating a term like log(x - c), the derivative is 1/(x - c), multiplied by the derivative of (x - c), which further simplifies the process of obtaining the overall derivative.

Domain Analysis

When a function includes a logarithmic term, it's essential to determine the domain of the function by ensuring that the argument of the logarithm remains positive. In any function involving log(x - c), the domain is restricted to values of x greater than c.

Derivative and Monotonicity

To assess where a function is increasing or decreasing, one computes its derivative and then examines the sign of the derivative over the function's domain. A function is increasing in an interval if its derivative is positive throughout that interval.

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