consider the function\n\n$f(x)=3x^{2}+5x - 2$\n\na) determine the intervals on which $f$ is concave up and…

consider the function\n\n$f(x)=3x^{2}+5x - 2$\n\na) determine the intervals on which $f$ is concave up and concave down.\n$f$ is concave up on:\n$f$ is concave down on:\n\nb) based on your answer to part (a), determine the inflection points of $f$. each point should be entered as an ordered pair (that is, in the form $(x,y)$).\n(separate multiple answers by commas.)\n\nnote: when using interval notation in webwork, remember that:\n you use inf for $\\infty$ and -inf for $-\\infty$,\n and use u for the union symbol.\nenter dne if an answer does not exist.\n\nnote: you can earn partial credit on this problem.

consider the function\n\n$f(x)=3x^{2}+5x - 2$\n\na) determine the intervals on which $f$ is concave up and concave down.\n$f$ is concave up on:\n$f$ is concave down on:\n\nb) based on your answer to part (a), determine the inflection points of $f$. each point should be entered as an ordered pair (that is, in the form $(x,y)$).\n(separate multiple answers by commas.)\n\nnote: when using interval notation in webwork, remember that:\n you use inf for $\\infty$ and -inf for $-\\infty$,\n and use u for the union symbol.\nenter dne if an answer does not exist.\n\nnote: you can earn partial credit on this problem.

Answer

Explanation:

Step1: Find the first derivative

Using the power rule ((x^n)^\prime = nx^{n - 1}), for (f(x)=3x^{2}+5x - 2), (f^\prime(x)=6x + 5).

Step2: Find the second derivative

Differentiate (f^\prime(x)) again. (f^{\prime\prime}(x)=(6x + 5)^\prime=6).

Since (f^{\prime\prime}(x)=6>0) for all (x\in(-\infty,\infty)):

  • For part (a), the function (f(x)) is concave - up on ((-\infty,\infty)) and there is no interval where it is concave - down.
  • For part (b), an inflection point occurs where (f^{\prime\prime}(x)) changes sign. Since (f^{\prime\prime}(x)) is a constant ((f^{\prime\prime}(x) = 6) never changes sign), there are no inflection points.

Answer:

a) (f) is concave up on: ((-\infty,\infty)) (f) is concave down on: DNE b) DNE