suppose that (f(x)=10 - 2e^{-x}). (a) list all critical numbers of (f). if there are no critical numbers…

suppose that (f(x)=10 - 2e^{-x}). (a) list all critical numbers of (f). if there are no critical numbers, enter none. (b) use interval notation to indicate where (f(x)) is increasing. increasing: (c) list the (x) - coordinates of all local maxima of (f). if there are no local maxima, enter none. (x) values of local maxima = (d) find the (x) - coordinates of all local minima of (f). if there are no local minima, enter none. (x) values of local minima = (e) use interval notation to indicate where (f(x)) is concave down. concave down: (f) list the (x) values of all inflection points of (f). if there are no inflection points, enter none. (x) values of inflection points = (g) list the horizontal asymptotes of (f). if there are no horizontal asymptotes, enter none. horizontal asymptotes (y=) (h) use all of the preceding information to sketch a graph of (f). when youre finished, enter a 1 in the box below. graph complete:

suppose that (f(x)=10 - 2e^{-x}). (a) list all critical numbers of (f). if there are no critical numbers, enter none. (b) use interval notation to indicate where (f(x)) is increasing. increasing: (c) list the (x) - coordinates of all local maxima of (f). if there are no local maxima, enter none. (x) values of local maxima = (d) find the (x) - coordinates of all local minima of (f). if there are no local minima, enter none. (x) values of local minima = (e) use interval notation to indicate where (f(x)) is concave down. concave down: (f) list the (x) values of all inflection points of (f). if there are no inflection points, enter none. (x) values of inflection points = (g) list the horizontal asymptotes of (f). if there are no horizontal asymptotes, enter none. horizontal asymptotes (y=) (h) use all of the preceding information to sketch a graph of (f). when youre finished, enter a 1 in the box below. graph complete:

Answer

  1. Find the derivative of (y = f(x)=10 - 2e^{-x}):
    • Recall the derivative of (e^{-x}) is (-e^{-x}). Using the sum - rule and constant - multiple rule of differentiation, if (y = 10-2e^{-x}), then (y^\prime=f^\prime(x)=2e^{-x}).
  2. (A) Find critical numbers:
    • Critical numbers occur where (f^\prime(x) = 0) or (f^\prime(x)) is undefined.
    • Set (f^\prime(x)=2e^{-x}=0). Since (e^{-x}=\frac{1}{e^{x}}) and (e^{x}>0) for all real (x), (2e^{-x}) is never zero. Also, (f^\prime(x)) is defined for all real (x).

    • Answer: NONE

  3. (B) Find where (f(x)) is increasing:
    • A function (y = f(x)) is increasing when (f^\prime(x)>0).
    • Since (f^\prime(x)=2e^{-x}=\frac{2}{e^{x}}>0) for all real (x) (because (e^{x}>0) for all (x\in R)).
    • In interval notation, the interval of increase is ((-\infty,\infty)).

    • Answer: ((-\infty,\infty))

  4. (C) Find local maxima and minima:
    • Since there are no critical numbers (where the derivative is zero or undefined), there are no local maxima or minima.

    • Answer: NONE

  5. (D) Find local minima:
    • As above, since there are no critical numbers, there are no local minima.

    • Answer: NONE

  6. (E) Find where (f(x)) is concave down:
    • First, find the second - derivative of (y = f(x)).
      • Since (f^\prime(x)=2e^{-x}), then (f^{\prime\prime}(x)=- 2e^{-x}).
      • A function (y = f(x)) is concave down when (f^{\prime\prime}(x)<0).
      • Set (f^{\prime\prime}(x)=-2e^{-x}<0). Since (e^{-x}=\frac{1}{e^{x}}>0) for all real (x), (-2e^{-x}<0) for all real (x).
      • In interval notation, the interval where (f(x)) is concave down is ((-\infty,\infty)).

      • Answer: ((-\infty,\infty))

  7. (F) Find inflection points:
    • Inflection points occur where (f^{\prime\prime}(x)) changes sign. Since (f^{\prime\prime}(x)=-2e^{-x}<0) for all real (x), (f^{\prime\prime}(x)) never changes sign.

    • Answer: NONE

  8. (G) Find horizontal asymptotes:
    • Calculate (\lim_{x\rightarrow\infty}f(x)=\lim_{x\rightarrow\infty}(10 - 2e^{-x})).
      • As (x\rightarrow\infty), (e^{-x}=\frac{1}{e^{x}}\rightarrow0). So, (\lim_{x\rightarrow\infty}(10 - 2e^{-x}) = 10).
      • Calculate (\lim_{x\rightarrow-\infty}f(x)=\lim_{x\rightarrow-\infty}(10 - 2e^{-x})). As (x\rightarrow-\infty), (e^{-x}\rightarrow\infty), and (\lim_{x\rightarrow-\infty}(10 - 2e^{-x})=-\infty).
      • The horizontal asymptote is (y = 10).

      • Answer: (y = 10)

  9. (H) Sketch the graph:
    • The function (y = 10-2e^{-x}) has a horizontal asymptote (y = 10), is increasing on ((-\infty,\infty)), is concave down on ((-\infty,\infty)), and has no critical numbers, local maxima, local minima, or inflection points. When (x = 0), (y=10 - 2=8). The graph starts below the horizontal asymptote (y = 10) (at (y = 8) when (x = 0)) and increases towards (y = 10) as (x\rightarrow\infty).

So, the answers are: (A) NONE (B) ((-\infty,\infty)) (C) NONE (D) NONE (E) ((-\infty,\infty)) (F) NONE (G) (y = 10) (H) (Sketch as described above)