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To determine over which interval the function [tex]\( f(x) = 6 \log_2(x) - 3 \)[/tex] is increasing at the greatest rate, we'll follow these steps:
1. Differentiate the function [tex]\( f(x) \)[/tex] to obtain the rate of change, i.e., the first derivative [tex]\( f'(x) \)[/tex].
We start by recalling the derivative of [tex]\( \log_2(x) \)[/tex]. The natural logarithm ln(x) and the log base 2 have a relationship expressed as:
[tex]\[ \log_2(x) = \frac{\ln(x)}{\ln(2)} \][/tex]
Therefore:
[tex]\[ \frac{d}{dx} \log_2(x) = \frac{1}{x \ln(2)} \][/tex]
2. Apply the chain rule to differentiate [tex]\( f(x) \)[/tex]:
[tex]\[ f(x) = 6 \log_2(x) - 3 \][/tex]
Differentiating each term separately, we get:
[tex]\[ f'(x) = 6 \cdot \frac{1}{x \ln(2)} = \frac{6}{x \ln(2)} \][/tex]
3. Analyze the first derivative [tex]\( f'(x) \)[/tex]:
We notice that [tex]\( \frac{6}{x \ln(2)} \)[/tex] is a decreasing function as [tex]\( x \)[/tex] increases because the denominator grows larger, reducing the overall value of the fraction. Therefore, [tex]\( f'(x) \)[/tex] is greater when [tex]\( x \)[/tex] is smaller, which means [tex]\( f(x) \)[/tex] increases at the greatest rate for smaller values of [tex]\( x \)[/tex].
4. Select an interval to determine where the rate of increase is the greatest:
Since [tex]\( f(x) \)[/tex] is defined for all [tex]\( x > 0 \)[/tex], let's choose an interval close to zero but not including zero (since log of zero is undefined).
We consider the interval [tex]\( (0, 1] \)[/tex], but to avoid undefined behavior when [tex]\( x \)[/tex] approaches zero, let’s pick a specific lower bound, say, [tex]\( x = 0.01 \)[/tex].
5. Evaluate [tex]\( f'(x) \)[/tex] at the boundaries of the interval [tex]\( x \)[/tex] in [tex]\( [0.01, 1] \)[/tex]:
For [tex]\( x = 0.01 \)[/tex]:
[tex]\[ f'(0.01) = \frac{6}{0.01 \ln(2)} \approx 865.617 \][/tex]
For [tex]\( x = 1 \)[/tex]:
[tex]\[ f'(1) = \frac{6}{1 \ln(2)} \approx 8.656 \][/tex]
In conclusion, the function [tex]\( f(x) = 6 \log_2(x) - 3 \)[/tex] increases at the greatest rate over the interval [tex]\( 0.01 < x \leq 1 \)[/tex]. The rate of increase is highest for [tex]\( x \)[/tex] close to 0.01 and decreases as [tex]\( x \)[/tex] approaches 1.
1. Differentiate the function [tex]\( f(x) \)[/tex] to obtain the rate of change, i.e., the first derivative [tex]\( f'(x) \)[/tex].
We start by recalling the derivative of [tex]\( \log_2(x) \)[/tex]. The natural logarithm ln(x) and the log base 2 have a relationship expressed as:
[tex]\[ \log_2(x) = \frac{\ln(x)}{\ln(2)} \][/tex]
Therefore:
[tex]\[ \frac{d}{dx} \log_2(x) = \frac{1}{x \ln(2)} \][/tex]
2. Apply the chain rule to differentiate [tex]\( f(x) \)[/tex]:
[tex]\[ f(x) = 6 \log_2(x) - 3 \][/tex]
Differentiating each term separately, we get:
[tex]\[ f'(x) = 6 \cdot \frac{1}{x \ln(2)} = \frac{6}{x \ln(2)} \][/tex]
3. Analyze the first derivative [tex]\( f'(x) \)[/tex]:
We notice that [tex]\( \frac{6}{x \ln(2)} \)[/tex] is a decreasing function as [tex]\( x \)[/tex] increases because the denominator grows larger, reducing the overall value of the fraction. Therefore, [tex]\( f'(x) \)[/tex] is greater when [tex]\( x \)[/tex] is smaller, which means [tex]\( f(x) \)[/tex] increases at the greatest rate for smaller values of [tex]\( x \)[/tex].
4. Select an interval to determine where the rate of increase is the greatest:
Since [tex]\( f(x) \)[/tex] is defined for all [tex]\( x > 0 \)[/tex], let's choose an interval close to zero but not including zero (since log of zero is undefined).
We consider the interval [tex]\( (0, 1] \)[/tex], but to avoid undefined behavior when [tex]\( x \)[/tex] approaches zero, let’s pick a specific lower bound, say, [tex]\( x = 0.01 \)[/tex].
5. Evaluate [tex]\( f'(x) \)[/tex] at the boundaries of the interval [tex]\( x \)[/tex] in [tex]\( [0.01, 1] \)[/tex]:
For [tex]\( x = 0.01 \)[/tex]:
[tex]\[ f'(0.01) = \frac{6}{0.01 \ln(2)} \approx 865.617 \][/tex]
For [tex]\( x = 1 \)[/tex]:
[tex]\[ f'(1) = \frac{6}{1 \ln(2)} \approx 8.656 \][/tex]
In conclusion, the function [tex]\( f(x) = 6 \log_2(x) - 3 \)[/tex] increases at the greatest rate over the interval [tex]\( 0.01 < x \leq 1 \)[/tex]. The rate of increase is highest for [tex]\( x \)[/tex] close to 0.01 and decreases as [tex]\( x \)[/tex] approaches 1.
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