consider the function\n\n f(x)=\frac{x^{4}}{4}-3 x^{3}-3 \n\na) determine the intervals on which ( f ) is…

consider the function\n\n f(x)=\frac{x^{4}}{4}-3 x^{3}-3 \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\nis, in the form ( (x, y) ) ).\n(separate multiple answers by commas.)\n\n(round to three decimal places at needed.)\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.

consider the function\n\n f(x)=\frac{x^{4}}{4}-3 x^{3}-3 \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\nis, in the form ( (x, y) ) ).\n(separate multiple answers by commas.)\n\n(round to three decimal places at needed.)\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.

Answer

Explanation:

Step1: Find the first and second derivatives

Given ( f(x)=\frac{x^{4}}{4}-3x^{3}-3 ). The first derivative ( f^{\prime}(x)=x^{3}-9x^{2}) (using the power rule ((x^{n})^\prime = nx^{n - 1})). The second derivative ( f^{\prime\prime}(x)=3x^{2}-18x=3x(x - 6)).

Step2: Find the critical points of ( f^{\prime\prime}(x))

Set ( f^{\prime\prime}(x)=0), so (3x(x - 6)=0). Solving (3x(x - 6)=0) gives (x = 0) and (x=6).

Step3: Test intervals for concavity

  • For the interval ((-\infty,0)), let (x=-1). Then (f^{\prime\prime}(-1)=3\times(-1)\times(-1 - 6)=21>0).
  • For the interval ((0,6)), let (x = 1). Then (f^{\prime\prime}(1)=3\times1\times(1 - 6)=-15<0).
  • For the interval ((6,\infty)), let (x = 7). Then (f^{\prime\prime}(7)=3\times7\times(7 - 6)=21>0).

Step4: Find (y) - values for inflection points

When (x = 0), (y=f(0)=\frac{0^{4}}{4}-3\times0^{3}-3=-3). When (x = 6), (y=f(6)=\frac{6^{4}}{4}-3\times6^{3}-3=\frac{1296}{4}-3\times216-3=324 - 648-3=-327).

Answer:

a) (f) is concave up on: ((-\infty,0)\cup(6,\infty)) (f) is concave down on: ((0,6)) b) ((0,-3),(6,-327))