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\nanswers as needed.)\n\nb. there are no local minima\n\nfind the open interval(s) on which the function is differentiable and is concave up. select the correct choice below and, if necessary, fill in the answer box to complete your choice.\n\na. the curve is concave up 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.)\n\nb. the curve is never concave up.

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\nanswers as needed.)\n\nb. there are no local minima\n\nfind the open interval(s) on which the function is differentiable and is concave up. select the correct choice below and, if necessary, fill in the answer box to complete your choice.\n\na. the curve is concave up 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.)\n\nb. the curve is never concave up.

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=\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) - 56x^{2}}{147(x^{2}-25)^{\frac{4}{3}}}\ &=\frac{28x^{2}-700 - 56x^{2}}{147(x^{2}-25)^{\frac{4}{3}}}\ &=\frac{- 28x^{2}-700}{147(x^{2}-25)^{\frac{4}{3}}}\ &=\frac{- 4(x^{2}+25)}{21(x^{2}-25)^{\frac{4}{3}}} \end{align*} ]

The function (y = \frac{3}{7}(x^{2}-25)^{\frac{2}{3}}) is differentiable for all (x\neq\pm5) (since the first derivative (y^\prime=\frac{4x}{7(x^{2}-25)^{\frac{1}{3}}}) is undefined at (x = \pm5)).

For concavity, we consider the sign of (y^{\prime\prime}). The denominator (21(x^{2}-25)^{\frac{4}{3}}>0) for (x\neq\pm5), and the numerator (-4(x^{2}+25)<0) for all real (x)

Answer:

B. The curve is never concave up.