Question

consider the function ( f(x)=8x + 5x^{-1} ).
note that this function has no inflection points, but ( f(x) ) is undefined at ( x = b ) where
( b=)
for each of the following intervals, tell whether ( f(x) ) is concave up (type in cu) or concave down (type in cd).
( (-infty,b):)
( (b,infty):)

Explanation:

Step1: Find the first - derivative

Using the power rule \( (x^n)^\prime=nx^{n - 1} \), for \( f(x)=8x + 5x^{-1} \), we have \( f^\prime(x)=8-5x^{-2}=8-\frac{5}{x^{2}} \).

Step2: Find the second - derivative

Differentiate \( f^\prime(x) \) with respect to \( x \). Using the power rule again, \( f^{\prime\prime}(x)=10x^{-3}=\frac{10}{x^{3}} \).

Step3: Find where \( f^{\prime\prime}(x) \) is undefined

Set the denominator of \( f^{\prime\prime}(x) \) equal to zero. For \( y = \frac{10}{x^{3}} \), when \( x = 0 \), \( f^{\prime\prime}(x) \) is undefined. So \( B = 0 \).

Step4: Test intervals for concavity

• For the interval \( (-\infty,0) \), let \( x=-1 \). Then \( f^{\prime\prime}(-1)=\frac{10}{(-1)^{3}}=-10<0 \). So \( f(x) \) is concave down (CD) on \( (-\infty,0) \).
• For the interval \( (0,\infty) \), let \( x = 1 \). Then \( f^{\prime\prime}(1)=\frac{10}{1^{3}}=10>0 \). So \( f(x) \) is concave up (CU) on \( (0,\infty) \).

Answer:

\( B = 0 \), \( (-\infty,B):CD \), \( (B,\infty):CU \)