3. for this problem, consider the function\nanswer the following questions and use your answers to sketch an…

3. for this problem, consider the function\nanswer the following questions and use your answers to sketch an accurate graph of ( f(x) ). no need to\nuse a calculator!\n(a) find ( f^{prime}(x) ).\n( f^{prime}(x)=)\n(b) find the critical points of ( f(x) ).\ncritical points:\n(c) find the interval(s) on which ( f(x) ) is increasing and the interval(s) on which ( f(x) ) is decreasing.\n(d) find ( f^{prime prime}(x) ).\n( f^{prime prime}(x)=)\n(e) find the interval(s) on which ( f(x) ) is concave up and the interval(s) on which ( f(x) ) is concave\nfown.\n(f) find any inflection points of ( f(x) ).\ninflection points:

3. for this problem, consider the function\nanswer the following questions and use your answers to sketch an accurate graph of ( f(x) ). no need to\nuse a calculator!\n(a) find ( f^{prime}(x) ).\n( f^{prime}(x)=)\n(b) find the critical points of ( f(x) ).\ncritical points:\n(c) find the interval(s) on which ( f(x) ) is increasing and the interval(s) on which ( f(x) ) is decreasing.\n(d) find ( f^{prime prime}(x) ).\n( f^{prime prime}(x)=)\n(e) find the interval(s) on which ( f(x) ) is concave up and the interval(s) on which ( f(x) ) is concave\nfown.\n(f) find any inflection points of ( f(x) ).\ninflection points:

Answer

Explanation:

Step1: Differentiate ( f(x) )

Given ( f(x)=\frac{1}{3}(x^{4}-4x^{3}) ), use the power rule ( (x^{n})^\prime = nx^{n - 1} ). ( f^\prime(x)=\frac{1}{3}(4x^{3}-12x^{2})=\frac{4}{3}x^{3}-4x^{2} )

Step2: Find critical points

Set ( f^\prime(x) = 0 ), so ( \frac{4}{3}x^{3}-4x^{2}=0 ). Factor out ( \frac{4}{3}x^{2} ): ( \frac{4}{3}x^{2}(x - 3)=0 ). Solutions are ( x = 0 ) and ( x=3 ) (critical points).

Step3: Differentiate ( f^\prime(x) )

Differentiate ( f^\prime(x)=\frac{4}{3}x^{3}-4x^{2} ) using the power rule. ( f^{\prime\prime}(x)=4x^{2}-8x=4x(x - 2) )

Step4: Find concavity intervals

Set ( f^{\prime\prime}(x)=0 ), ( 4x(x - 2)=0 ), solutions ( x = 0,x = 2 ). Test intervals:

  • For ( x\lt0 ), let ( x=-1 ), ( f^{\prime\prime}(-1)=4\times(-1)\times(-1 - 2)=12\gt0 ), concave - up on ( (-\infty,0) ).
  • For ( 0\lt x\lt2 ), let ( x = 1 ), ( f^{\prime\prime}(1)=4\times1\times(1 - 2)=-4\lt0 ), concave - down on ( (0,2) ).
  • For ( x\gt2 ), let ( x = 3 ), ( f^{\prime\prime}(3)=4\times3\times(3 - 2)=12\gt0 ), concave - up on ( (2,\infty) ).

Step5: Find inflection points

Since concavity changes at ( x = 0 ) and ( x = 2 ). ( f(0)=\frac{1}{3}(0^{4}-4\times0^{3}) = 0 ), ( f(2)=\frac{1}{3}(2^{4}-4\times2^{3})=\frac{1}{3}(16 - 32)=-\frac{16}{3} ). Inflection points are ( (0,0) ) and ( (2,-\frac{16}{3}) ).

Answer:

(a) ( f^\prime(x)=\frac{4}{3}x^{3}-4x^{2} ) (b) Critical points: ( x = 0,x = 3 ) (c) Increasing on ( (3,\infty) ), decreasing on ( (-\infty,3) ) (d) ( f^{\prime\prime}(x)=4x^{2}-8x ) (e) Concave up on ( (-\infty,0)\cup(2,\infty) ), concave down on ( (0,2) ) (f) Inflection points: ( (0,0),(2,-\frac{16}{3}) )