identify the inflection points and local maxima and minima of the function below and its associated graph…

identify the inflection points and local maxima and minima of the function below and its associated graph. identify the open intervals on which the function is differentiable and is concave up and concave down.\n\n( y=\frac{3}{7}left(x^{2}-25\right)^{\frac{2}{3}} )\n\nfind the open interval(s) on which the function is differentiable and is concave down. select the correct choice below and, if necessary, fill in the answer box to complete your choice.\n\na. the curve is concave down on the open interval(s) (simplify your answer. type your answer in interval notation. type an exact answer, using radicals as needed. use a comma to separate answers as needed )\nb. the curve is never concave down.

identify the inflection points and local maxima and minima of the function below and its associated graph. identify the open intervals on which the function is differentiable and is concave up and concave down.\n\n( y=\frac{3}{7}left(x^{2}-25\right)^{\frac{2}{3}} )\n\nfind the open interval(s) on which the function is differentiable and is concave down. select the correct choice below and, if necessary, fill in the answer box to complete your choice.\n\na. the curve is concave down on the open interval(s) (simplify your answer. type your answer in interval notation. type an exact answer, using radicals as needed. use a comma to separate answers as needed )\nb. the curve is never concave down.

Answer

Explanation:

Step1: Find the first derivative

Using the chain rule, if (y = \frac{3}{7}(x^{2}-25)^{\frac{2}{3}}), then (y^\prime=\frac{3}{7}\times\frac{2}{3}(x^{2}-25)^{-\frac{1}{3}}\times2x=\frac{4x}{7(x^{2}-25)^{\frac{1}{3}}})

Step2: Find the second derivative

Using the quotient rule ((\frac{u}{v})^\prime=\frac{u^\prime v - uv^\prime}{v^{2}}), where (u = 4x), (u^\prime=4), (v = 7(x^{2}-25)^{\frac{1}{3}}), (v^\prime=7\times\frac{1}{3}(x^{2}-25)^{-\frac{2}{3}}\times2x=\frac{14x}{3}(x^{2}-25)^{-\frac{2}{3}})

[ \begin{align*} y^{\prime\prime}&=\frac{4\times7(x^{2}-25)^{\frac{1}{3}}-4x\times\frac{14x}{3}(x^{2}-25)^{-\frac{2}{3}}}{49(x^{2}-25)^{\frac{2}{3}}}\ &=\frac{28(x^{2}-25)-\frac{56x^{2}}{3}}{49(x^{2}-25)^{\frac{4}{3}}}\ &=\frac{\frac{84(x^{2}-25)- 56x^{2}}{3}}{49(x^{2}-25)^{\frac{4}{3}}}\ &=\frac{84x^{2}-2100 - 56x^{2}}{147(x^{2}-25)^{\frac{4}{3}}}\ &=\frac{28x^{2}-2100}{147(x^{2}-25)^{\frac{4}{3}}}\ &=\frac{4(x^{2}-75)}{21(x^{2}-25)^{\frac{4}{3}}} \end{align*} ]

Step3: Determine where (y^{\prime\prime}<0)

Set (y^{\prime\prime}<0), (\frac{4(x^{2}-75)}{21(x^{2}-25)^{\frac{4}{3}}}<0)

Since ((x^{2}-25)^{\frac{4}{3}}>0) for (x\neq\pm5), we solve (x^{2}-75<0)

(x^{2}-75=(x - 5\sqrt{3})(x + 5\sqrt{3})<0)

The solution of (x^{2}-75<0) is (-5\sqrt{3}<x<5\sqrt{3}), and the function is differentiable for (x\neq\pm5)

Answer:

A. The curve is concave down on the open interval(s) ((-5\sqrt{3},5\sqrt{3}))