Calculus Help Rolle\'s theorem. Mean value theorem. Max Points to the person tha

Question

Calculus Help Rolle's theorem. Mean value theorem. Max Points to the person that gives me COMPLETE and CORRECT solutions to the problems. STEP BY STEP please.

1. Determine whether Rolle's Theorem applies to the following function on the given interval. If so, nd all
points guaranteed to exist by Rolle's Theorem.
f(x) = x(x - 1)^2; [0; 1]

2. Determine whether the Mean Value Theorem applies to the following function on the given interval. If
so, nd all points guaranteed to exist by the Mean Value Theorem.
g(x) = |x| ; [1; 4]

3. Evaluate: lim x-->0 (3sin(4x))/(5x)

4. Evaluate: lim x-->Infinity   (6x^4-x^2)/(3x^4+12)

5. Evaluate: lim x-->0 -xcsc(x)

Explanation / Answer

1. f(x) = x(x - 1)^2?

f is differentiable at [0,1] , and f(0) = f(1) = 0

so there exists c shch that f'(c) = 0

so f'(x) = 2x(x-1) + (x-1)^2

f'(c) = 2c(c - 1) + (c - 1)^2 = 0

so (c - 1) (2c + c - 1) = 0

so either c = 1 (refused)

or 3c - 1 = 0

so c = 1/3

__________________________________________________

2. g(x) = |x|

at [1 , 4] , g(x) = x is differentiable at ]1 , 4[

so there is c such that g'(c) = [g(4) - g(1)] / (4 - 1) = 3/3 = 1

this implies that c is any point at the open interval ]1 , 4 [

______________________________________________________

3. lim x-->0 (3sin(4x))/(5x)

= (3/5) lim x-->0 (sin(4x)) / x )

= (3/5) * 4 lim x-->0 (sin(4x)) / (4x) )

= 12/5 * 1

= 12/5

_____________________________________________________

4. lim x-->Infinity   (6x^4-x^2)/(3x^4+12)

= lim x -> inf x^4 [6 - (1/x^2)] / x^4 [3 + (12/x^4)]

= 6 - 0 / 3 + 0

= 6/3 = 2

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5. lim x-->0 -xcsc(x) = lim -x / sinx

= - lim x-->0 (x /sinx)

= - 1 / lim x ->0 (sinx /x)

= -1 / 1

= - 1

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