Question

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:

Explanation:

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 \(y^\prime=f^\prime(x)=2e^{-x}\).

1. (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

1. (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: \((-\infty,\infty)\)

1. (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

1. (D) Find local minima:

• As above, since there are no critical numbers, there are no local minima.
• # Answer: NONE

1. (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: \((-\infty,\infty)\)

1. (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

1. (G) Find horizontal asymptotes:

• Calculate \(\lim_{x

ightarrow\infty}f(x)=\lim_{x
ightarrow\infty}(10 - 2e^{-x})\).

• As \(x

ightarrow\infty\), \(e^{-x}=\frac{1}{e^{x}}
ightarrow0\). So, \(\lim_{x
ightarrow\infty}(10 - 2e^{-x}) = 10\).

• Calculate \(\lim_{x

ightarrow-\infty}f(x)=\lim_{x
ightarrow-\infty}(10 - 2e^{-x})\). As \(x
ightarrow-\infty\), \(e^{-x}
ightarrow\infty\), and \(\lim_{x
ightarrow-\infty}(10 - 2e^{-x})=-\infty\).

• The horizontal asymptote is \(y = 10\).
• # Answer: \(y = 10\)

1. (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

ightarrow\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)

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 \(y^\prime=f^\prime(x)=2e^{-x}\).

1. (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

1. (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: \((-\infty,\infty)\)

1. (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

1. (D) Find local minima:

• As above, since there are no critical numbers, there are no local minima.
• # Answer: NONE

1. (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: \((-\infty,\infty)\)

1. (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

1. (G) Find horizontal asymptotes:

• Calculate \(\lim_{x

ightarrow\infty}f(x)=\lim_{x
ightarrow\infty}(10 - 2e^{-x})\).

• As \(x

ightarrow\infty\), \(e^{-x}=\frac{1}{e^{x}}
ightarrow0\). So, \(\lim_{x
ightarrow\infty}(10 - 2e^{-x}) = 10\).

• Calculate \(\lim_{x

ightarrow-\infty}f(x)=\lim_{x
ightarrow-\infty}(10 - 2e^{-x})\). As \(x
ightarrow-\infty\), \(e^{-x}
ightarrow\infty\), and \(\lim_{x
ightarrow-\infty}(10 - 2e^{-x})=-\infty\).

• The horizontal asymptote is \(y = 10\).
• # Answer: \(y = 10\)

1. (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

ightarrow\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)