Explanation Definition used: First derivative test: Suppose c to be a critical number for a function f . Assume that f is continuous on ( a , b ) and differentiable on ( a , b ) except at c , and c is the only critical number of f in ( a , b ) . f ( c ) is a relative maximum of f at c if f ′ ( x ) > 0 in the interval ( a , c ) and f ′ ( x ) < 0 in the interval ( c , b ) . f ( c ) is a relative minimum f at c if f ′ ( x ) < 0 in the interval ( a , c ) and f ′ ( x ) > 0 in the interval ( c , b ) . Relative maximum: A function is said to have a relative maximum f ( c ) at c if f ′ ( x ) > 0 in the interval ( a , c ) and f ′ ( x ) < 0 in the interval ( c , b ) . Relative minimum: A function is said to have a relative maximum f ( c ) at c if f ′ ( x ) < 0 in the interval ( a , c ) and f ′ ( x ) > 0 in the interval ( c , b ) . Calculation: The degree of change is given by the function D ( x ) = − x 4 + 8 x 3 + 80 x 2 . Obtain the derivative of the function as follows. D ′ ( x ) = ( − x 4 + 8 x 3 + 80 x 2 ) ′ = − ( 4 x 3 ) + 8 ( 3 x 2 ) + 80 ( 2 x ) = − 4 x 3 + 24 x 2 + 160 x Set D ′ ( c ) = 0 and find the value of c . − 4 c 3 + 24 c 2 + 160 c = 0 − 4 c ( c 2 − 6 c − 40 ) = 0 c ( c 2 − 10 c + 4 c − 40 ) = 0 c ( c ( c − 10 ) + 4 ( c − 10 ) ) = 0 Simplify further as follows

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Chapter 13.2, Problem 57E

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To find: The amount of discrepancy at which the maximum change in attitude in the audience is received where the degree of change is given by the function D(x)=−x4+8x3+80x2 .

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Chapter 13.2, Problem 56E

Chapter 13.2, Problem 58E

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The amount of discrepancy at which the maximum change in attitude in the audience is received where the degree of change is given by the function D ( x ) = − x 4 + 8 x 3 + 80 x 2 .

Solution Summary: The author explains that the amount of discrepancy at which the maximum change in attitude in the audience is received is 10.

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To find: The amount of discrepancy at which the maximum change in attitude in the audience is received where the degree of change is given by the function D(x)=−x4+8x3+80x2 .

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