find the x and y coordinates of all inflection points.\n\n f(x) = 4 x ^ { \frac { 5 } { 3 } } + 3 \n\nwhat…

find the x and y coordinates of all inflection points.\n\n f(x) = 4 x ^ { \frac { 5 } { 3 } } + 3 \n\nwhat is/are the inflection point(s)? select the correct choice below and, if necessary, fill in the answer box to complete your choice.\n\na. the inflection point(s) is/are\n(type an ordered pair. use a comma to separate answers as needed.)\nb. there are no inflection points.

find the x and y coordinates of all inflection points.\n\n f(x) = 4 x ^ { \frac { 5 } { 3 } } + 3 \n\nwhat is/are the inflection point(s)? select the correct choice below and, if necessary, fill in the answer box to complete your choice.\n\na. the inflection point(s) is/are\n(type an ordered pair. use a comma to separate answers as needed.)\nb. there are no inflection points.

Answer

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\neq0). 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))