Math Problem Statement
[ f(x)=\frac{x}{x^{2}+15 x+50} ]a) Give the domain of ( f ) (in interval notation) ( \square )b) Find the critical numbers of ( f ). ( \square ) (Separate multiple answers by commas.)c) Determine the intervals on which ( f ) is increasing and decreasing.( f ) is increasing on: ( \square ) and( f ) is decreasing on: ( \square ) . ( \square ) , and ( \square )d) Use the First Derivative Test to determine whether each critical point is a local maximum, minimum, or neither.Local maxima occur at ( x= ) ( \square ) (Separate multiple answers by commas.)Local minima occur at ( x= ) ( \square ) (Separate multiple answers by commas.)
Solution
Let's solve this step by step:
Function Definition:
(a) Domain of
The function is undefined where the denominator equals 0. Solve: Factorize: Thus, the denominator is 0 at and . Therefore, the domain is:
(b) Critical Numbers of
To find the critical numbers, compute the derivative using the quotient rule: Simplify the numerator: Thus:
Set to find critical points:
Critical numbers are:
(c) Intervals of Increase and Decrease
The critical points and points where the function is undefined divide the number line into intervals:
Test the sign of in each interval by choosing a test point:
-
In : Choose
-
In : Choose
-
In : Choose
-
In : Choose
-
In : Choose
From this, is:
- Increasing on
- Decreasing on
(d) Local Maxima and Minima (First Derivative Test)
- At : changes from positive to negative → local maximum.
- At : changes from positive to negative → local maximum.
- At : changes from negative to positive → local minimum.
Thus:
- Local maxima:
- Local minima:
Final Answers
(a) Domain:
(b) Critical numbers:
(c) Intervals:
(d) Local maxima: Local minima:
Would you like further clarification or a graph of to visualize these results?
5 Related Questions:
- How do we apply the second derivative test for concavity analysis?
- Can have absolute extrema on its domain? Why or why not?
- How do vertical asymptotes affect the behavior of ?
- What happens to as or ?
- How do the critical points influence the shape of ?
Tip: When analyzing functions, always start with the domain to avoid undefined operations!
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Math Problem Analysis
Mathematical Concepts
Functions
Critical Points
Intervals of Increase and Decrease
First Derivative Test
Formulas
Quotient Rule for Derivatives: f'(x) = (g(x)h'(x) - g'(x)h(x)) / (h(x))^2
Factoring Quadratic Equations
Theorems
First Derivative Test
Suitable Grade Level
Grades 11-12
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