the questions in level 1 are introductory problems. the hints contain links to videos covering this content…

the questions in level 1 are introductory problems. the hints contain links to videos covering this content. question 4 (1 point) retake question let ( f(x)=x^{3 / 5}(x - 3) ). the inflection point(s) of ( f(x) ) is(are) (choose all that apply): ( x = 0 ) ( x = 3 ) ( x=\frac{9}{8} ) this function does not have any inflection points. ( x=\frac{-3}{4} ) view hint for question 4 question 5 (1 point) retake question

the questions in level 1 are introductory problems. the hints contain links to videos covering this content. question 4 (1 point) retake question let ( f(x)=x^{3 / 5}(x - 3) ). the inflection point(s) of ( f(x) ) is(are) (choose all that apply): ( x = 0 ) ( x = 3 ) ( x=\frac{9}{8} ) this function does not have any inflection points. ( x=\frac{-3}{4} ) view hint for question 4 question 5 (1 point) retake question

Answer

Explanation:

Step1: Expand the function

First, expand (f(x)=x^{3/5}(x - 3)=x^{8/5}-3x^{3/5}).

Step2: Find the first - derivative

Using the power rule ((x^n)^\prime=nx^{n - 1}), (f^\prime(x)=\frac{8}{5}x^{3/5}-\frac{9}{5}x^{-2/5}=\frac{8x - 9}{5x^{2/5}}).

Step3: Find the second - derivative

Using the quotient rule ((\frac{u}{v})^\prime=\frac{u^\prime v - uv^\prime}{v^{2}}), where (u = 8x - 9), (u^\prime=8), (v = 5x^{2/5}), (v^\prime=2x^{-3/5}).

(f^{\prime\prime}(x)=\frac{8\times5x^{2/5}-(8x - 9)\times2x^{-3/5}}{25x^{4/5}}=\frac{40x^{2/5}-16x^{2/5}+18x^{-3/5}}{25x^{4/5}}=\frac{24x + 18}{25x^{7/5}}).

Step4: Find the inflection points

Set (f^{\prime\prime}(x)=0), then (24x+18 = 0), (x=-\frac{3}{4}). But we also need to check the domain of (f^{\prime\prime}(x)). The function (y = f(x)) is defined for all real (x), and (f^{\prime\prime}(x)) changes sign at (x = 0).

When (x\lt0), let (x=-1), (f^{\prime\prime}(-1)=\frac{24\times(-1)+18}{25\times(-1)^{7/5}}=\frac{- 6}{-25}=\frac{6}{25}\gt0).

When (0\lt x\lt-\frac{3}{4}), let (x =-\frac{1}{2}), (f^{\prime\prime}(-\frac{1}{2})=\frac{24\times(-\frac{1}{2})+18}{25\times(-\frac{1}{2})^{7/5}}\lt0) (since the numerator (24\times(-\frac{1}{2})+18=-12 + 18=6) and the denominator (25\times(-\frac{1}{2})^{7/5}\lt0)).

When (x\gt-\frac{3}{4}), let (x = 0.1), (f^{\prime\prime}(0.1)=\frac{24\times0.1+18}{25\times(0.1)^{7/5}}\gt0).

Answer:

(x = 0)