Question

review a: (unit 2: topics 2.3, 2.10 – 2.11) exp and log graphs name:

1. the graph of the function ( f ) is shown above. which of the following statements about ( f ) is correct?

(a) ( f ) is increasing at an increasing rate.
(b) ( f ) is increasing at a decreasing rate.
(c) ( f ) is decreasing at an increasing rate.
(d) ( f ) is decreasing at a decreasing rate.

1. the graph of the function ( g ) is shown above. which of the following statements about ( g ) is correct?

(a) the function ( g ) is increasing and the graph of ( g ) is concave up.
(b) the function ( g ) is increasing and the graph of ( g ) is concave down.
(c) the function ( g ) is decreasing and the graph of ( g ) is concave up.
(d) the function ( g ) is decreasing and the graph of ( g ) is concave down.

1. the logarithmic function ( h ) is defined by ( h(x) = -2 ln x ). which of the following statements about ( h ) is correct?

(a) the function ( h ) is increasing and the graph of ( h ) is concave up.
(b) the function ( h ) is increasing and the graph of ( h ) is concave down.
(c) the function ( h ) is decreasing and the graph of ( h ) is concave up.
(d) the function ( h ) is decreasing and the graph of ( h ) is concave down.

Explanation:

Question 1

Step1: Analyze the graph of \( f \)

The graph of \( f \) is a curve that is decreasing (as \( x \) increases, \( y \) decreases) and the slope (rate of change) is becoming less negative (the curve is flattening out), which means it's decreasing at a decreasing rate.

Step2: Evaluate the options

• Option A: \( f \) is not increasing, so A is wrong.
• Option B: \( f \) is not increasing, so B is wrong.
• Option C: \( f \) is decreasing, but the rate of decrease is decreasing (not increasing), so C is wrong.
• Option D: Matches the analysis of the graph of \( f \).

Step1: Analyze the graph of \( g \)

The graph of \( g \) is rising as \( x \) increases (so \( g \) is increasing) and the curve is concave up (it's curving upwards, like a cup).

Step2: Evaluate the options

• Option A: Matches the analysis (increasing and concave up).
• Option B: \( g \) is not concave down, so B is wrong.
• Option C: \( g \) is not decreasing, so C is wrong.
• Option D: \( g \) is not decreasing, so D is wrong.

Step1: Analyze the function \( h(x)= - 2\ln x \)

First, find the derivative to check if it's increasing or decreasing. The derivative \( h^\prime(x)=-\frac{2}{x} \). For \( x>0 \) (domain of \( \ln x \)), \( h^\prime(x)<0 \), so \( h \) is decreasing. Then find the second derivative: \( h^{\prime\prime}(x)=\frac{2}{x^{2}} \). For \( x > 0 \), \( h^{\prime\prime}(x)>0 \), so the graph is concave up.

Step2: Evaluate the options

• Option A: \( h \) is not increasing, so A is wrong.
• Option B: \( h \) is not increasing, so B is wrong.
• Option C: Matches the analysis (decreasing and concave up).
• Option D: \( h \) is concave up, not down, so D is wrong.

Answer:

D. \( f \) is decreasing at a decreasing rate.

Question 2