The interval (s) of decrease of the function f(x)= x^2 log27-6 x log27+(3 x^2-18 x+24) log(x^2-6

step 1 in this problem to find increase or decrease of the function we first find the value of f dash o

The interval (s) of decrease of the function f(x)= x^2...

The interval $(s)$ of decrease of the function $f(x)=$ $x^{2} \log 27-6 x \log 27+\left(3 x^{2}-18 x+24\right) \log \left(x^{2}-6 x\right.$
$+8)$ is
(a) $(3-\sqrt{1+13 e}, 2)$
(b) $(4,3+\sqrt{1+1 / 3 e})$
(c) $(3,4+\sqrt{1+1 / 3 e})$
(d) none of these

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Key Concepts

Domain Restrictions

When dealing with functions involving logarithms, it is crucial to consider the domain restrictions since the argument of a logarithmic function must be positive. This affects the valid range of x-values when finding intervals of monotonic behavior.

Inequality Solving

Once the derivative is computed, solving the inequality where the derivative is negative (f?'(x) < 0) will reveal the interval(s) where the function is decreasing. This involves algebraic manipulation and careful handling of inequalities.

Derivative

The derivative of a function represents its instantaneous rate of change. In problems asking for intervals of increase or decrease, the derivative is computed to determine where the function's slope is positive (increasing) or negative (decreasing).

First Derivative Test

The first derivative test involves setting the derivative greater than or less than zero to determine the intervals where a function is increasing or decreasing, respectively. This is a fundamental tool in analyzing the behavior of functions.

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