Given f'(x) = (4x^2+5x+1)/(x-8)^(1/3) sketch f(x) where f(x) is a continuous function   we are given the function (4x^2+5x+1)/(x-8)^(1/3)  which is the first derivative of some continuous function f(x). a) Show the work needed to determine the intervals where f(x) is increasing and decreasing, and find the x-coordinates of all the local maximum and minimum points.   b) Draw a sketch of some continuous function f(x) tht has the properties you found in part a.

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Let the derivative function be called `f'(x)`

First of all, notice that there is a singularity at `x=8` because here we are dividing by zero.

Next, factorise the top of the function giving

`f'(x) = (4x^2 + 5x+1)/(x-8)^(1/3) = ((4x+1)(x+1))/(x-8)^(1/3)`

This tells us that `f(x)` has turning points/points of inflection at `x=-1/4``x=-1` because the gradient of `f(x)``f'(x)`, is zero there.

Examine these points to determine whether they are maxima/minima/turning points.

To do this we examine the sign of the second derivative `f''(x) = d/(dx) f'(x)`

Now `f''(x) = d/(dx) (4x^2 + 5x+1)/(x-8)^(1/3) = (8x+5)/(x-8)^(1/3) -(1/3)(1)(4x^2+5x+1)/(x-8)^(-2/3)`

`= (8x+5)/(x-8)^(1/3) -(1/3)((4x+1)(x+1))/(x-8)^(-2/3)`

At `x=-1/4`` ``f''(x) approx (5-2)/(-8)^(1/3) = -3/2`  which is negative implying there is a maximum at `x=-1/4`

At `x=-1``f''(x) approx (5-8)/(-8)^(1/3) = 3/2`  which is positive implying there is a minimum at `x=-1`

Now look at what is happening around the value `x=8`

When `x=7``f'(x) = (4(49)+5(7)+1)/(-1)^(1/3)`  which is negative

When `x=9``f'(x) = (4(81)+5(9)+1)/1^(1/3)`  which is positive

Therefore as `x-> 8` from below, `f(x) -> -oo` and as `x-> 8` from above, `f(x) -> oo`

There is a minimum at `x=-1`, a maximum at `x=-1/4`

The gradient is negative over `(-oo,-1)`, positive over `(-1,-1/4)`, negative over `(-1/4,8)` and positive over `(8,oo)`

A graph` ` `g(x)` that has these properties is

`g(x) = (4/3x^3 + 5/2x^2 +x)/(x-8)` , `x < 8`

`= -1/(x-8)` , `x>8`



Minimum at -1, maximum at -1/4

Decreasing over (-inf,-1), increasing over (-1,-1/4),decreasing over (-1/4,8), increasing over (8,inf)


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