consider the following function. use a graphing utility to confirm your answers for parts (a) through (c)…

consider the following function. use a graphing utility to confirm your answers for parts (a) through (c). (if an answer does not exist, enter dne.)\n\n$f(x)=-4x^{2}+24x + 9$\n\n(a) find the critical numbers of f. (enter your answers as a comma-separated list.)\n\n$x=$\n\n(b) find the open intervals on which the function is increasing or decreasing. (enter your answers using interval notation.)\n\nincreasing\n\ndecreasing\n\n(c) apply the first derivative test to identify all relative extrema.\n\nrelative maximum $(x,y)=()$\n\nrelative minimum $(x,y)=()$

consider the following function. use a graphing utility to confirm your answers for parts (a) through (c). (if an answer does not exist, enter dne.)\n\n$f(x)=-4x^{2}+24x + 9$\n\n(a) find the critical numbers of f. (enter your answers as a comma-separated list.)\n\n$x=$\n\n(b) find the open intervals on which the function is increasing or decreasing. (enter your answers using interval notation.)\n\nincreasing\n\ndecreasing\n\n(c) apply the first derivative test to identify all relative extrema.\n\nrelative maximum $(x,y)=()$\n\nrelative minimum $(x,y)=()$

Answer

Explanation:

Step1: Find the derivative of (f(x))

The function is (f(x)=-4x^{2}+24x + 9). Using the power rule ((x^{n})^\prime=nx^{n - 1}), we have (f^\prime(x)=-8x+24).

Step2: Find the critical numbers

Set (f^\prime(x) = 0). So, (-8x + 24=0). Solving for (x): [ \begin{align*} -8x&=-24\ x&=3 \end{align*} ]

Step3: Determine the intervals of increase and decrease

  • For the interval ((-\infty,3)), let's test (x = 0). Then (f^\prime(0)=-8\times0 + 24=24>0). So the function is increasing on ((-\infty,3)).
  • For the interval ((3,\infty)), let's test (x = 4). Then (f^\prime(4)=-8\times4+24=-8<0). So the function is decreasing on ((3,\infty)).

Step4: Apply the First - Derivative Test for relative extrema

Since (f(x)) changes from increasing ((f^\prime(x)>0)) to decreasing ((f^\prime(x)<0)) at (x = 3). Find (y) - value: (y=f(3)=-4\times3^{2}+24\times3 + 9=-36 + 72+9=45). There is a relative maximum at ((3,45)) and no relative minimum (because the function is a parabola opening downwards ((a=-4<0)) and has only one critical point).

Answer:

(a) (x = 3) (b) Increasing: ((-\infty,3)); Decreasing: ((3,\infty)) (c) Relative maximum: ((x,y)=(3,45)); Relative minimum: DNE