Question

find the x and y coordinates of all inflection points.

f(x) = 4 x ^ { \frac { 5 } { 3 } } + 3

what is/are the inflection point(s)? select the correct choice below and, if necessary, fill in the answer box to complete your choice.

a. the inflection point(s) is/are
(type an ordered pair. use a comma to separate answers as needed.)
b. there are no inflection points.

Explanation:

Step1: Find the first derivative

Using the power rule \( (x^n)^\prime=nx^{n - 1} \), for \( y = 4x^{\frac{5}{3}}+3 \), the first derivative \( y^\prime=f^\prime(x)=4\times\frac{5}{3}x^{\frac{5}{3}-1}=\frac{20}{3}x^{\frac{2}{3}} \).

Step2: Find the second derivative

Again, using the power rule. \( y^{\prime\prime}=f^{\prime\prime}(x)=\frac{20}{3}\times\frac{2}{3}x^{\frac{2}{3}-1}=\frac{40}{9}x^{-\frac{1}{3}}=\frac{40}{9x^{\frac{1}{3}}} \).

Step3: Analyze the second - derivative for inflection points

Inflection points occur where \( f^{\prime\prime}(x) = 0 \) or \( f^{\prime\prime}(x) \) is undefined.
Set \( f^{\prime\prime}(x)=0 \), \(\frac{40}{9x^{\frac{1}{3}}}=0\), this equation has no solution since the numerator \(40
eq0\).
The second - derivative \( f^{\prime\prime}(x)=\frac{40}{9x^{\frac{1}{3}}}\) is undefined when \(x = 0\).

Step4: Find the \(y\) - coordinate when \(x = 0\)

Substitute \(x = 0\) into the original function \(y=f(x)=4x^{\frac{5}{3}}+3\). Then \(y=4\times0^{\frac{5}{3}}+3=3\).

Answer:

A. The inflection point(s) is/are \((0,3)\)