hw 16 - first derivative test section 3.2: problem 3\n(1 point)\nconsider the function\n$f(x)=2 + x…

hw 16 - first derivative test section 3.2: problem 3\n(1 point)\nconsider the function\n$f(x)=2 + x - x^{2}$.\n1. $f$ is increasing on the intervals\n2. $f$ is decreasing on the intervals\n3. the relative maxima of $f$ occur at $x=$\n4. the relative minima of $f$ occur at $x=$\nin the last two type the word \none\ if there is not a relative maximum or minimum.\nnote: you can earn partial credit on this problem.\npreview my answers submit answers

hw 16 - first derivative test section 3.2: problem 3\n(1 point)\nconsider the function\n$f(x)=2 + x - x^{2}$.\n1. $f$ is increasing on the intervals\n2. $f$ is decreasing on the intervals\n3. the relative maxima of $f$ occur at $x=$\n4. the relative minima of $f$ occur at $x=$\nin the last two type the word \none\ if there is not a relative maximum or minimum.\nnote: you can earn partial credit on this problem.\npreview my answers submit answers

Answer

Explanation:

Step1: Find the first derivative

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

Step2: Find the critical points

Set (f^\prime(x) = 0), so (1-2x=0). Solving for (x) gives (x=\frac{1}{2}).

Step3: Determine the intervals of increase and decrease

  • Choose a test - point in the interval ((-\infty,\frac{1}{2})), say (x = 0). Then (f^\prime(0)=1-2\times0=1>0). So (f(x)) is increasing on ((-\infty,\frac{1}{2})).
  • Choose a test - point in the interval ((\frac{1}{2},\infty)), say (x = 1). Then (f^\prime(1)=1-2\times1=-1<0). So (f(x)) is decreasing on ((\frac{1}{2},\infty)).

Step4: Find relative maxima and minima

Since (f(x)) changes from increasing to decreasing at (x=\frac{1}{2}), by the first - derivative test, (f(x)) has a relative maximum at (x=\frac{1}{2}) and no relative minimum (because the function only changes from increasing to decreasing, not the other way around).

Answer:

  1. ((-\infty,\frac{1}{2}))
  2. ((\frac{1}{2},\infty))
  3. (\frac{1}{2})
  4. none