Explanation The formula to calculate the differential of a function h ( x ) = [ g ( x ) f ( x ) ] by Product Rule is, h ′ ( x ) = f ( x ) g ′ ( x ) + f ′ ( x ) g ( x ) The function f ( x ) increases in the interval ( a , b ) , if f ′ ( x ) > 0 for every x in the interval ( a , b ) . The function g ( x ) increases in the interval ( a , b ) , if g ′ ( x ) > 0 for every x in the interval ( a , b ) . Assume two positive functions f ( x ) and g ( x ) that is f ( x ) > 0 and g ( x ) > 0 . Assume a function h ( x ) be product of two functions f ( x ) and g ( x ) , h ( x ) = ( f g ) ( x ) (1) Differentiate both sides with respect to x in equation (1). d d x ( h ( x ) ) = d d x [ ( f g ) ( x ) ] = g ( x ) d d x [ f ( x ) ] + f ( x ) d d x [

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Chapter 4.1, Problem 100E

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To check: Whether the statement “If f(x) and g(x) are positive on (a,b) and both f and g increases on (a,b) , then fg increases on (a,b) .” is true or false.

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Chapter 4.1, Problem 99E

Chapter 4.1, Problem 101E

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Ch. 4.1 - Explain each of the following: a. f is increasing...Ch. 4.1 - Prob. 2CQ Ch. 4.1 - Prob. 3CQ Ch. 4.1 - Prob. 4CQ Ch. 4.1 - Prob. 5CQ Ch. 4.1 - Prob. 1E Ch. 4.1 - Prob. 2E Ch. 4.1 - Prob. 3E Ch. 4.1 - Prob. 4E Ch. 4.1 - Prob. 5E

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To check: Whether the statement “If f ( x ) and g ( x ) are positive on ( a , b ) and both f and g increases on ( a , b ) , then f g increases on ( a , b ) .” is true or false.

Solution Summary: The author explains the formula to calculate the differential of a function h(x) by Product Rule.

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To check: Whether the statement “If f(x) and g(x) are positive on (a,b) and both f and g increases on (a,b) , then fg increases on (a,b) .” is true or false.

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