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 given function is f ( x ) = ln ( 3 x ) 2 x 2 . Obtain the derivative of the function as follows. f ′ ( x ) = ( ln ( 3 x ) 2 x 2 ) ′ = ( 2 x 2 ) ( ln ( 3 x ) ) ′ − ( ln ( 3 x ) ) ( 2 x 2 ) ′ ( 2 x 2 ) 2 = ( 2 x 2 ) 3 3 x − ( ln ( 3 x ) ) ⋅ 4 x 4 x 4 = 2 x − 4 x ln ( 3 x ) 4 x 4 Simplify further as follows. f ′ ( x ) = 2 x − 4 x ln ( 3 x ) 4 x 4 = 2 x ( 1 − 2 ln ( 3 x ) ) 4 x 4 = 1 − 2 ln ( 3 x ) 2 x 3 Clearly, at x = 0 , the derivative does not exist and it cannot be a critical number as the function is not defined at this point

Finite Mathematics and Calculus with Applications (10th Edition)

10th Edition

ISBN: 9780321979407

Author: Margaret L. Lial, Raymond N. Greenwell, Nathan P. Ritchey

Publisher: PEARSON

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Chapter 13, Problem 32RE

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To find: The x-values of all points where the function f(x)=ln(3x)2x2 has relative maxima and minima, and to find the values of all relative extrema.

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Chapter 13, Problem 31RE

Chapter 13, Problem 33RE

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A box with a square base and open top must have a volume of 171500 cm3cm3. We wish to find the dimensions of the box that minimize the amount of material used.First, find a formula for the surface area of the box in terms of only xx, the length of one side of the square base.[Hint: use the volume formula to express the height of the box in terms of xx.]Simplify your formula as much as possible.A(x)=A(x)= Next, find the derivative, A'(x)A′(x).A'(x)=A′(x)= Now, calculate when the derivative equals zero, that is, when A'(x)=0A′(x)=0. [Hint: multiply both sides by x2x2.]A'(x)=0A′(x)=0 when x=x=We next have to make sure that this value of xx gives a minimum value for the surface area. Let's use the second derivative test. Find A"(x)(x).A"(x)=(x)= Evaluate A"(x)(x) at the xx-value you gave above.NOTE: Since your last answer is positive, this means that the graph of A(x)A(x) is concave up around that value, so the zero of A'(x)A′(x) must indicate a local minimum for A(x)A(x). (Your boss is…

Find the amount s of advertising (in thousands of dollars) that maximizes the profit P (in thousands of dollars). x = _______ Find the point of diminishing returns. (s, P) = (_____)

A box with a square base and open top must have a volume of 37044 cm3cm3. We wish to find the dimensions of the box that minimize the amount of material used.First, find a formula for the surface area of the box in terms of only xx, the length of one side of the square base.[Hint: use the volume formula to express the height of the box in terms of xx.]Simplify your formula as much as possible.A(x)=A(x)= Next, find the derivative, A'(x)A′(x).A'(x)=A′(x)= Now, calculate when the derivative equals zero, that is, when A'(x)=0A′(x)=0. [Hint: multiply both sides by x2x2.]A'(x)=0A′(x)=0 when x=x=We next have to make sure that this value of xx gives a minimum value for the surface area. Let's use the second derivative test. Find A"(x)(x).A"(x)=(x)= Evaluate A"(x)(x) at the xx-value you gave above.

Chapter 13 Solutions

Finite Mathematics and Calculus with Applications (10th Edition)

Ch. 13.1 - Find where the function is increasing and...Ch. 13.1 - Prob. 2YT Ch. 13.1 - Prob. 3YT Ch. 13.1 - Prob. 4YT Ch. 13.1 - Prob. 1WE Ch. 13.1 - Prob. 2WE Ch. 13.1 - Prob. 3WE Ch. 13.1 - Prob. 4WE Ch. 13.1 - Prob. 5WE Ch. 13.1 - Prob. 6WE

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The x -values of all points where the function f ( x ) = ln ( 3 x ) 2 x 2 has relative maxima and minima , and to find the values of all relative extrema.

Solution Summary: The author explains that the relative maximum of 0.83 occurs at x=0.55.

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To find: The x-values of all points where the function f(x)=ln(3x)2x2 has relative maxima and minima, and to find the values of all relative extrema.

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Chapter 13 Solutions