   Chapter 3.R, Problem 19E

Chapter
Section
Textbook Problem

# 17-28 Use the guidelines of Section 3.5 to sketch the curve. y = 3 x 4 − 4 x 3 + 2

To determine

To Sketch:

The curve y = 3x4-4x3+2

Explanation

1) Concept:

i) A domain is the set of x values that satisfy the function.

ii) To find x-intercept, put =0, and to find y-intercept, put x=0 in the given function.

iii) Symmetry: To find symmetry, replace x by –x and check the behavior of function. Thus, if f-x=fx, then it is an even function, so it has y axis symmetry, and if f-x=-fx then it is an odd function, so it has x axis symmetry. And if f-x-fxf(x) then it has no symmetry.

iv) An asymptote is a tangent at infinity. To find horizontal, vertical and slant asymptotes, follow the rules.

v) The function is increasing if f'x>0  and decreasing if f'x<0 in that particular interval.

vi) The number f(c) has a local maximum value of f  if fcf(x) when x is near c. and a local minimum value of f if fc f(x) when x is near c.

vii) If f''x>0 the function is concave up and f''x<0 the function is concave down in that particular interval. And f''x=0 gives the values of inflection points

2) Given:

The function y = 3x4-4x3+2

3) Calculation:

The given function is y = 3x4-4x3+2

A) Domain

Since y=3x4-4x3+2  is a polynomial, its domain is -,  .

B) Intercepts

For y intercept plug x=0  in the given function and solve it

y=3(0)4-40+2=2

y intercept is (0, 2)

Now for x-intercept set y=0 then equation becomes

0=3x4-4x3+2

By using the graphing calculator, x takes complex values

Therefore, there are no x- intercepts

C) Symmetry

For symmetry replace x by (-x), therefore,

f-x=3-x4-4-x3+2=3x4+4x3+2

Since, fxf-x or f-x-f(x)

That means there is no symmetry

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