Explanation Given: The function is given as: f ( x ) = 2 x + 8 x Graph: To plot the graph, the steps to be followed in TI-83 are as follows: Step 1: In the TI-83 graphical calculator, press the Y = key. Step 2: Enter the equation of f ( x ) as 2 X + 8 / X in Y 1 . Step 3: Press the TRACE and Y key. Step 4: Go to WINDOW and set X min = − 15 , X max = 15 , Y min = − 15 , Y max = 15 . Step 5: Click the TRACE key to obtain the graph. The graph is obtained as follows: The x - and y- intercepts are calculated as follows: Since x = 0 is not in the domain of the function because f ( x ) is not defined at that point. So there is no y -intercept. To find the x -intercepts, solve f ( x ) = 0 . 2 x + 8 x = 0 2 x 2 + 8 x = 0 2 x 2 + 8 = 0 x 2 = − 4 Since x 2 is never negative, thus there is no x- intercept. Power rule of differentiation states that if f ( x ) = x n for any number n , then f ′ ( x ) = n x n − 1 . Addition rule of differentiation states that if: f ( x ) = u ( x ) + v ( x ) Then, f ′ ( x ) is given by: f ′ ( x ) = u ′ ( x ) + v ′ ( x ) To find all the critical points, calculate the first-order derivative of the given function. Use the addition rule and power rule: f ′ ( x ) = d d x ( 2 x + 8 x ) = d d x ( 2 x ) + d d x ( 8 x ) = 2 − 8 x 2 Now put f ′ ( x ) = 0 to find the critical points. 2 − 8 x 2 = 0 x 2 = 4 x = ± 2 Now find the second-order derivative of the function: f ″ ( x ) = d f ( x ) d x Use the addition rule and power rule: f ″ ( x ) = d d x ( 2 − 8 x 2 ) = d d x ( 2 ) − d d x ( 8 x 2 ) = 0 + 16 x 3 = 16 x 3 Since there is no value of x such that f ″ ( x ) is 0

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ISBN: 9780135335215

Author: Lial

Publisher: PEARSON

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Chapter 12.6, Problem 14E

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To graph: The function f(x)=2x+8x and locate the point of inflection and local extrema.

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Chapter 12.6, Problem 13E

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To graph: The function f ( x ) = 2 x + 8 x and locate the point of inflection and local extrema.

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To graph: The function f(x)=2x+8x and locate the point of inflection and local extrema.

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