Choose the values of a and b for which the function f(x) = ab^x meets the given requirement that f is decreasing for its entire domain.
Understand the Problem
The question asks for the values of 'a' and 'b' such that the function f(x) = ab^x is decreasing over its entire domain. The options provided involve inequalities and equalities for 'a' and 'b'.
Answer
The values are \( a < 0 \) and \( b > 1 \) or \( 0 < b < 1 \) and \( a > 0 \).
Answer for screen readers
The values of ( a ) and ( b ) such that the function ( f(x) = ab^x ) is decreasing are ( a < 0 ) and ( b > 1 ) or ( a > 0 ) and ( 0 < b < 1 ).
Steps to Solve
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Understand the Function Behavior To determine when the function ( f(x) = ab^x ) is decreasing, we analyze its derivative. A function is decreasing when its derivative is negative.
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Find the Derivative The derivative of the function is given by the product of the constant ( a ) and the derivative of ( b^x ): $$ f'(x) = a \frac{d}{dx}(b^x) = a b^x \ln(b) $$
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Set Up Inequality for Decreasing Behavior For ( f(x) ) to be decreasing, we set the derivative less than zero: $$ a b^x \ln(b) < 0 $$
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Analyze Conditions for a and b
- If ( b > 1 ): then ( \ln(b) > 0 ) and we require ( a < 0 ) for the product to be negative.
- If ( 0 < b < 1 ): then ( \ln(b) < 0 ) and we require ( a > 0 ) to make the product negative.
- Combine Cases We conclude:
- For ( b > 1 ): ( a < 0 )
- For ( 0 < b < 1 ): ( a > 0 )
Thus ( a < 0 ) and ( b > 1 ) can also satisfy the conditions. Therefore, the function is decreasing for ( a < 0 ) and ( 0 < b < 1 ).
The values of ( a ) and ( b ) such that the function ( f(x) = ab^x ) is decreasing are ( a < 0 ) and ( b > 1 ) or ( a > 0 ) and ( 0 < b < 1 ).
More Information
In exponential functions, the growth or decay rate depends heavily on the values of the base ( b ) and the coefficient ( a ). When ( b > 1 ) represents growth, choosing ( a < 0 ) flips the function to descending. If ( 0 < b < 1 ), the function naturally decreases, but ( a ) must be positive for the output to remain negative.
Tips
- A common mistake is confusing the criteria for ( b ): remembering that ( b > 1 ) leads to growth and must be paired with a negative ( a ).
- Misunderstanding logarithmic properties can lead to incorrect signs in inequalities.
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