Explanation The given equation is: y = 4 x 2 − x Differentiate the function with respect to the x . d d x ( y ) = d d x ( 4 x 2 − x ) f ′ ( x ) = 4 × ( − 2 ) x 2 + 1 − 1 = − 8 x 3 − 1 Second derivative of the function y = 4 x 2 − x . f ″ ( x ) = d d x ( − 8 x 3 − 1 ) f ″ ( x ) = − 8 × ( − 3 ) x 3 + 1 = 24 x 4 For the point of inflection f ″ ( x ) = 0 . 24 x 4 = 0 x = ∞ Hence, at x = ∞ the curve changes its concavity. Substitute, x = ∞ in the function y = 4 x 2 − x . y = 4 ∞ 2 − ∞ = − ∞ Hence, point of inflection is ( ∞ , − ∞ ) . For, local maxima and local minima. Substitute, f ′ ( x ) = 0 in f ′ ( x ) = − 8 x 3 − 1 = 0 . − 8 x 3 − 1 = 0 − 8 − x 3 = 0 x 3 + 8 = 0 x 3 + 2 3 = 0 ( x + 2 ) = 0 and ( x 2 − 2 x + 4 ) = 0 So x = − 2 and the next equation gives an imaginary value. Hence, Local maxima or local minima occurs when x = − 2 . Substitute, x = − 2 in f ″ ( x ) = 24 x 4 . f ″ ( x ) = 24 x 4 f ″ ( − 2 ) = 24 ( − 2 ) 4 = 1.5 f ″ ( 0 ) > 0 Substitute, x = − 2 in y = 4 x 2 − x . y = 4 − 2 2 − ( − 2 ) = 1 + 2 = 3 Hence, at ( − 2 , 3 ) _ local minima occur. Slope of the function f ( x ) = 4 x 2 − x . For, f ′ ( x ) < 0 . 4 x 2 − x < 0 ( 4 − x 3 ) < 0 On further simplification, x lies between ( 1.587 , − 0.794 ) . Hence, slope of the function f ( x ) = 4 x 2 − x is decreasing when − ∞ < x < − 2 _ . For, f ′ ( x ) > 0 . 4 x 2 − x > 0 ( 4 − x 3 ) > 0 On further simplification, x lies between ( 1.587 , − 0.794 ) . Hence, slope of the function f ( x ) = 4 x 2 − x is increasing when − 2 < x < 0 _ . Concavity of the function y = 4 x 2 − x . For f ″ ( x ) < 0 . 24 x 4 < 0 24 < x 4 x 4 > 24 x > 2

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Author: Washington

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Rate of Change

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The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as:

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Chapter 24, Problem 8PT

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Chapter 24 Solutions

Basic Technical Mathematics

Ch. 24.1 - For the parabola y = 4 − x2, at the point (3, −5)...Ch. 24.1 - Prob. 2PE Ch. 24.1 - Prob. 1E Ch. 24.1 - Prob. 2E Ch. 24.1 - Prob. 3E Ch. 24.1 - Prob. 4E Ch. 24.1 - Prob. 5E Ch. 24.1 - Prob. 6E Ch. 24.1 - Prob. 7E Ch. 24.1 - Prob. 8E

Ch. 24.1 - Prob. 9E Ch. 24.1 - Prob. 10E Ch. 24.1 - Prob. 11E Ch. 24.1 - Prob. 12E Ch. 24.1 - In Exercises 11–14, find the equations of the...Ch. 24.1 - Prob. 14E Ch. 24.1 - Prob. 15E Ch. 24.1 - Prob. 16E Ch. 24.1 - Prob. 17E Ch. 24.1 - Prob. 18E Ch. 24.1 - Prob. 19E Ch. 24.1 - Prob. 20E Ch. 24.1 - Prob. 21E Ch. 24.1 - Where does the normal line to the parabola y = x —...Ch. 24.1 - Prob. 23E Ch. 24.1 - Prob. 24E Ch. 24.1 - A certain suspension cable with supports on the...Ch. 24.1 - Prob. 26E Ch. 24.1 - Prob. 27E Ch. 24.1 - Prob. 28E Ch. 24.1 - Prob. 29E Ch. 24.1 - Prob. 30E Ch. 24.2 - In Example 1, let x1 = 0.3, and find x2. 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A microprocessor chip is being designed with a...Ch. 24.7 - Prob. 18E Ch. 24.7 - What are the dimensions of the largest rectangular...Ch. 24.7 - A rectangular storage area is to be constructed...Ch. 24.7 - Prob. 21E Ch. 24.7 - Prob. 22E Ch. 24.7 - Prob. 23E Ch. 24.7 - Prob. 24E Ch. 24.7 - Prob. 25E Ch. 24.7 - Prob. 26E Ch. 24.7 - Prob. 27E Ch. 24.7 - Prob. 28E Ch. 24.7 - Prob. 29E Ch. 24.7 - Prob. 30E Ch. 24.7 - Prob. 31E Ch. 24.7 - Prob. 32E Ch. 24.7 - Prob. 33E Ch. 24.7 - What is the minimum slope of the curve y = x5 −...Ch. 24.7 - Prob. 35E Ch. 24.7 - Prob. 36E Ch. 24.7 - Prob. 37E Ch. 24.7 - Prob. 38E Ch. 24.7 - Prob. 39E Ch. 24.7 - Prob. 40E Ch. 24.7 - Prob. 41E Ch. 24.7 - Computer simulation shows that the drag F (in N)...Ch. 24.7 - Prob. 43E Ch. 24.7 - The potential energy E of an electric charge q due...Ch. 24.7 - An open box is to be made from a square piece of...Ch. 24.7 - Prob. 46E Ch. 24.7 - Prob. 47E Ch. 24.7 - Prob. 48E Ch. 24.7 - An oil pipeline is to be built from a refinery to...Ch. 24.7 - 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The graph of y = 4 x 2 − x .

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Chapter 24 Solutions