Explanation Definition used: An interior point c of the domain of f at which f ′ ( c ) = 0 or f ′ ( c ) fails to exist is called a critical point of f . The x -intercept is the point where the graph of the line crosses the x -axis. The y -intercept is the point where the graph of the line crosses the y -axis. An asymptote of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the x or y coordinates tends to infinity. Theorem used: Suppose f ″ exists on an open interval I . 1 If f ″ > 0 on I, then f is concave up on I . 2. If f ″ < 0 on I, then f is concave down on I . 3. If c is a point of I at which f ″ changes sign at c (from positive to negative, or vice versa), then f has an inflection point at c . Assume f is continuous on an interval that contains a critical point c , and assume f is differentiable on an interval containing c , except perhaps at c itself. 1. If f ′ changes sign from positive to negative as x increases through c , then f has a local maximum at c . 2. If f ′ changes sign from negative to positive as x increases through c , then f has a local minimum at c . 3. If f ′ is positive on both sides near c or negative on both sides near c , then f has no local extreme value at c . Result used: Graphing guidelines 1. Identify the domain or interval of interest. 2. Exploit symmetry. 3. Find the first and second derivatives. 4. Find critical points and possible inflection points. 5. Find intervals on which the function is increasing/decreasing and concave up/down. 6. Identify extreme values and inflection points. 7. Locate all asymptotes and determine end behaviour. 8. Find the intercepts. 9. Choose an appropriate graphing window and plot a graph. Calculation: Consider the function f ( x ) = 10 x 2 x 2 + 3 . Clearly, the given function is a polynomial function and so the domain is ( − ∞ , ∞ ) and the function has no asymptotes. Obtain the value of f ( − x ) as follows. f ( − x ) = 10 ( − x ) 2 ( − x ) 2 + 3 = 10 x 2 x 2 + 3 Hence, the function is symmetric about the origin. Compute the value of lim x → ∞ f ( x ) . lim x → ∞ f ( x ) = lim x → ∞ 10 x 2 x 2 + 3 = lim x → ∞ 10 x 2 x 2 x 2 + 3 x 2 = lim x → ∞ 10 1 + 3 x 2 = 10 Hence, the horizontal asymptote of f is y = 10 . Compute the first derivative of f as follows. f ′ ( x ) = d d x ( 10 x 2 x 2 + 3 ) = ( x 2 + 3 ) d d x ( 10 x 2 ) − ( 10 x 2 ) d d x ( x 2 + 3 ) ( x 2 + 3 ) 2 [ By quotient rule ] = ( x 2 + 3 ) ( 20 x ) − ( 10 x 2 ) ( 2 x ) ( x 2 + 3 ) 2 = 60 x ( x 2 + 3 ) 2 Compute the first derivative of f as follows. f ″ ( x ) = d d x ( 60 x ( x 2 + 3 ) 2 ) = ( x 2 + 3 ) 2 d d x ( 60 x ) − ( 60 x ) d d x ( ( x 2 + 3 ) 2 ) ( x 2 + 3 ) 4 [ By quotient rule ] = 60 ( x 2 + 3 ) 2 − ( 60 x ) ( 4 x ( x 2 + 3 ) ) ( x 2 + 3 ) 4 = 180 ( 1 − x ) ( 1 + x ) ( x 2 + 3 ) 3 Set f ′ ( x ) = 0 and simplify as follows. 60 x ( x 2 + 3 ) 2 = 0 x = 0 Hence, the critical points is x = 0 . Set f ″ ( x ) = 0 and simplify as follows. 180 ( 1 − x ) ( 1 + x ) ( x 2 + 3 ) 3 = 0 x = ± 1 Substitute x = 0 in f ″ ( x ) = 180 ( 1 − x ) ( 1 + x ) ( x 2 + 3 ) 3 . f ″ ( 0 ) = 180 ( 1 − 0 ) ( 1 + 0 ) ( 0 2 + 3 ) 3 = 180 ( 1 ) ( 1 ) ( 3 ) 3 = 180 27 > 0 Substitute x = − 2 in f ″ ( x ) = 180 ( 1 − x ) ( 1 + x ) ( x 2 + 3 ) 3

Calculus, Single Variable: Early Transcendentals (3rd Edition)

3rd Edition

ISBN: 9780134766850

Author: William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz

Publisher: PEARSON

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Chapter 4, Problem 31RE

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To graph: The function f(x)=10x2x2+3.

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Chapter 4, Problem 30RE

Chapter 4, Problem 32RE

Chapter 4 Solutions

Calculus, Single Variable: Early Transcendentals (3rd Edition)

Ch. 4.1 - Sketch the graph of a function that is continuous...Ch. 4.1 - Consider the function f(x) = x3. Where is the...Ch. 4.1 - Sketch the graph of a function that is continuous...Ch. 4.1 - Sketch the graph of a function that has an...Ch. 4.1 - What is a critical point of a function?Ch. 4.1 - Sketch the graph of a function f that has a local...Ch. 4.1 - Sketch the graph of a function f that has a local...Ch. 4.1 - Absolute maximum/minimum values Use the following...Ch. 4.1 - Prob. 12E Ch. 4.1 - Absolute maximum/minimum values Use the following...

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To graph: The function f ( x ) = 10 x 2 x 2 + 3 .

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