If 4\leq x g(x)\leq2x^4-2x^2+4 for all x, evaluate \lim_{x\to1}g(x)

urca3m403 2021-11-14 Answered
If 4xg(x)2x42x2+4 for all x, evaluate limx1g(x)
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Expert Answer

Drood1980
Answered 2021-11-15 Author has 16 answers
Given,
4xg(x)2x42x2+4 for all x.
Now,
limx14x=4(1)
=4
and
limx1(2x42x2+4)=2(1)42(1)2+4
=22+4
=4
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How to find the general antiderivative of f ( x ) = x ( 6 x ) 2 ?
I want to find the general antiderivative of f ( x ) = x ( 6 x ) 2 . However, I keep getting it wrong.
I am new to antiderivatives, but I think the first thing I should do is differentiate.
d d x ( 6 x ) 2 = 2 ( 6 x ) 1 = 2 ( 6 x )
f ( x ) = [ ( 6 x ) 2 1 ] + [ 2 ( 6 x ) x ] = ( 6 x ) 2 2 x ( 6 x )
According to the antiderivative power rule, when n 1, x n d x = x n + 1 n + 1 + C.
So it seems like ( 6 x ) 2 d x = ( 6 x ) 3 3 + C.
However, I can't find a rule that seems like it would work with 2 x ( 6 x ). The closest thing that I can find is the "Multiplication by Constant Rule", c f ( x ) d x = c ( f ( x ) ) d x, but I'm not sure if I'm allowed to change 2 x ( 6 x ) into the form 2 ( 6 x x 2 ).
2 ( 6 x x 2 )
= 2 ( 6 x x 2 )
= 2 ( 6 x x 2 )
= 2 ( 6 x 2 2 x 3 3 ) + C
But ( 6 x ) 3 3 ( 6 x 2 2 x 3 3 ) + C is incorrect.

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