summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of ( y =…

summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of ( y = f(x) ).\n\n( f(x)=4 x^{6}-9 x^{5} )\n\nthe function ( f ) has a local maximum at ( x = )\n(type an exact answer, using radicals as needed. use a comma to separate answers as needed.)\n\nthe function ( f ) has no local maximum.\n\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\n\nthe function ( f ) has a local minimum at ( x=\frac{15}{8} )\n(type an exact answer, using radicals as needed. type an integer or a fraction. use a comma to separate answers as needed.)\n\nthe function ( f ) has no local minimum.\n\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\n\na. the graph of ( f ) is concave upward on the subinterval(s)\n(type your answer in interval notation. type an exact answer, using radicals as needed. type an integer or a fraction. use a comma to separate answers as needed.)\n\nb. the function ( f ) is never concave upward

summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of ( y = f(x) ).\n\n( f(x)=4 x^{6}-9 x^{5} )\n\nthe function ( f ) has a local maximum at ( x = )\n(type an exact answer, using radicals as needed. use a comma to separate answers as needed.)\n\nthe function ( f ) has no local maximum.\n\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\n\nthe function ( f ) has a local minimum at ( x=\frac{15}{8} )\n(type an exact answer, using radicals as needed. type an integer or a fraction. use a comma to separate answers as needed.)\n\nthe function ( f ) has no local minimum.\n\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\n\na. the graph of ( f ) is concave upward on the subinterval(s)\n(type your answer in interval notation. type an exact answer, using radicals as needed. type an integer or a fraction. use a comma to separate answers as needed.)\n\nb. the function ( f ) is never concave upward

Answer

Explanation:

Step1: Find the first derivative

Use the power rule ((x^n)^\prime=nx^{n - 1}). For (y = f(x)=4x^{6}-9x^{5}), the first derivative (f^\prime(x)=24x^{5}-45x^{4}=3x^{4}(8x - 15)). Set (f^\prime(x)=0), then (3x^{4}(8x - 15)=0). Solving (3x^{4}(8x - 15)=0) gives (x = 0) or (x=\frac{15}{8}). Use the first - derivative test.

  • When (x\lt\frac{15}{8}(x\neq0)), let (x = 1), then (f^\prime(1)=24\times1^{5}-45\times1^{4}=24 - 45=-21\lt0).
  • When (x\gt\frac{15}{8}), let (x = 2), then (f^\prime(2)=24\times2^{5}-45\times2^{4}=24\times32-45\times16=768 - 720 = 48\gt0). Since the function changes from decreasing ((f^\prime(x)\lt0)) to increasing ((f^\prime(x)\gt0)) at (x=\frac{15}{8}), (x = \frac{15}{8}) is a local minimum. Since (f^\prime(x)) does not change sign at (x = 0) (the sign of (f^\prime(x)) is negative on both sides of (x = 0) for (x\neq0)), there is no local maximum.

Step2: Find the second derivative

Differentiate (f^\prime(x)=24x^{5}-45x^{4}) using the power rule. (f^{\prime\prime}(x)=120x^{4}-180x^{3}=60x^{3}(2x - 3)). Set (f^{\prime\prime}(x)=0), then (60x^{3}(2x - 3)=0). Solving (60x^{3}(2x - 3)=0) gives (x = 0) or (x=\frac{3}{2}).

  • When (x\lt0), let (x=-1), then (f^{\prime\prime}(-1)=120\times(- 1)^{4}-180\times(-1)^{3}=120 + 180=300\gt0).
  • When (0\lt x\lt\frac{3}{2}), let (x = 1), then (f^{\prime\prime}(1)=120\times1^{4}-180\times1^{3}=120 - 180=-60\lt0).
  • When (x\gt\frac{3}{2}), let (x = 2), then (f^{\prime\prime}(2)=120\times2^{4}-180\times2^{3}=120\times16-180\times8=1920 - 1440 = 480\gt0).

Answer:

The function (f) has no local maximum. The function (f) has a local minimum at (x=\frac{15}{8}). The graph of (f) is concave upward on the sub - intervals ((-\infty,0)\cup(\frac{3}{2},\infty))