Question

find the open intervals where the function is concave upward or concave downward. find any inflection points.

$f(x)=\frac{5}{x - 1}$

where is the function concave upward and where is it concave downward? select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice.

a. the function is concave downward on the interval(s) $\square$. the function is never concave upward.

(use integers or fractions for any numbers in the expression. use a comma to separate answers as needed.)

b. the function is concave upward on the interval(s) $\square$. the function is never concave downward.

(type your answer in interval notation. use integers or fractions for any numbers in the expression. use a comma to separate answers as needed.)

c. the function is concave upward on the interval(s) $\square$. the function is concave downward on the interval(s) $\square$.

(type your answers in interval notation. use integers or fractions for any numbers in the expressions. use a comma to separate answers as needed.)

d. the function is neither concave downward nor concave upward.

find any inflection points of $f$. select the correct choice below and, if necessary, fill in the answer box to complete your choice.

a. the function $f$ has an inflection point at $\square$.

(simplify your answer. type an ordered pair, using integers or fractions. use a comma to separate answers as needed.)

b. the function $f$ has no inflection points.

Explanation:

Step1: Find the first - derivative

Using the quotient rule \((\frac{u}{v})^\prime=\frac{u^\prime v - uv^\prime}{v^{2}}\), where \(u = 5\), \(u^\prime=0\), \(v=x - 1\), \(v^\prime = 1\).

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Step2: Find the second - derivative

Using the quotient rule again, where \(u=-5\), \(u^\prime = 0\), \(v=(x - 1)^{2}\), \(v^\prime=2(x - 1)\)

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Step3: Analyze concavity

Set \(f^{\prime\prime}(x)=0\), \(\frac{10}{(x - 1)^{3}} = 0\) has no solution.
The function \(y = f(x)\) is not defined at \(x = 1\).

• For \(x\lt1\), let \(x=0\), then \(f^{\prime\prime}(0)=\frac{10}{(0 - 1)^{3}}=-10\lt0\), so the function is concave downward on \((-\infty,1)\)
• For \(x\gt1\), let \(x = 2\), then \(f^{\prime\prime}(2)=\frac{10}{(2 - 1)^{3}}=10\gt0\), so the function is concave upward on \((1,\infty)\)

Step4: Analyze inflection points

Since \(x = 1\) is not in the domain of \(f(x)\) (because \(f(x)=\frac{5}{x - 1}\) is undefined at \(x = 1\)), there are no inflection points.

Answer:

• Concave upward: \((1,\infty)\)
• Concave downward: \((-\infty,1)\)
• Inflection points: The function \(f\) has no inflection points. So for the multiple - choice:
• For concave upward: C. The function is concave upward on the interval \((1,\infty)\)
• For concave downward: B. The function is concave downward on the interval \((-\infty,1)\)
• For inflection points: B. The function \(f\) has no inflection points.