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Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516

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BuyFindarrow_forward

Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516
Textbook Problem

Analyzing the Graph of a Function In Exercises 5-34, analyze and sketch a graph of the function. Label any intercepts, relative extrema, points of inflection, and asymptotes. Use a graphing utility to verify your results.

y = 2 x 4 + 3 x 2

To determine

To graph: The function y=2x4+3x2

Explanation

Given: The function y=2x4+3x2.

Graph: The function, being a polynomial has a domain of all real numbers. Thus the domain of the function would be (,).

The function is a polynomial and thus the range of the function would be (,98].

Now find the x and y intercepts by equating y and x to zero respectively to obtain:

The x-intercepts would be (62,0),(0,0),(62,0) and the y-intercept would be (0,0).

The function has no vertical or horizontal asymptotes

Now, differentiate the function with respect to x and equate it to zero to obtain the critical points.

f'(x)=2x(4x23)2x(4x23)=0x=0,32,32

This gives four test intervals (,32),(32,0),(0,32)(32,).

Let 1(,32).

f'(1)=2(1)(4(1)23)=2>0

The function is increasing on this interval.

Let 0.5(32,0).

f'(0.5)=2(0.5)(4(0.5)23)=2<0

The function is decreasing on this interval.

Let 0.5(0,32).

f'(0.5)=2(0.5)(4(0.5)23)=2>0

The function is increasing on this interval.

Let 1(32,).

f'(1)=2(1)(4(1)23)=2<0

The function is increasing on this interval.

Now obtain the value of the function at the critical points.

f(0)=2(0)4+3(0)2=0f(32)=2(32)4+3(32)2=98

And,

f(32)=2(32)4+3(32)2=98

Thus, the critical points are (32,98),(32,98)

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