question 14\ndetermine all intervals and shape of concavity of the graph of the function: $f(x)=7 +…

question 14\ndetermine all intervals and shape of concavity of the graph of the function: $f(x)=7 + 4x^{1/3}$.\nno correct answer choice is given.\nconcave down on $(0,\\infty)$, concave upon $(-\\infty,0)$,\nconcave up on $(0,\\infty)$, concave down on $(-\\infty,0)$,\nconcave up on $(0,\\infty)$.\nconcave down on $(0,\\infty)$

question 14\ndetermine all intervals and shape of concavity of the graph of the function: $f(x)=7 + 4x^{1/3}$.\nno correct answer choice is given.\nconcave down on $(0,\\infty)$, concave upon $(-\\infty,0)$,\nconcave up on $(0,\\infty)$, concave down on $(-\\infty,0)$,\nconcave up on $(0,\\infty)$.\nconcave down on $(0,\\infty)$

Answer

Explanation:

Step1: Find the first - derivative

Use the power rule ( (x^n)^\prime=nx^{n - 1}). For (y = f(x)=7 + 4x^{\frac{1}{3}}), then (y^\prime=f^\prime(x)=\frac{4}{3}x^{-\frac{2}{3}}=\frac{4}{3x^{\frac{2}{3}}}).

Step2: Find the second - derivative

Differentiate (y^\prime=\frac{4}{3}x^{-\frac{2}{3}}) with respect to (x) using the power rule again. (y^{\prime\prime}=f^{\prime\prime}(x)=-\frac{8}{9}x^{-\frac{5}{3}}=-\frac{8}{9x^{\frac{5}{3}}}).

Step3: Analyze the sign of the second - derivative

  • When (x>0), (y^{\prime\prime}=-\frac{8}{9x^{\frac{5}{3}}}<0).
  • When (x < 0), (y^{\prime\prime}=-\frac{8}{9x^{\frac{5}{3}}}>0) (since (x^{\frac{5}{3}}=x\cdot x^{\frac{2}{3}}) and (x<0,x^{\frac{2}{3}}>0)).

Answer:

Concave down on ((0,\infty)), concave up on ((-\infty,0))