Question 1191568
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Graph Approach:


If you typed "graph -x^3 + 5x^2 - 8x + 4" into google without quotes, then it will do as asked.
Though the usual tool I use is GeoGebra. Another handy grapher is Desmos.
There are tons of free options online.
If you prefer your handheld graphing calculator, then of course go for that option.


Whichever graphing tool you use, a cubic curve results which looks like a sort of "S" shape in a sense.


The curve crosses the x axis at x = 1 and x = 2
As the graph shows, when {{{1 < x < 2}}}, the curve is below the x axis.
Also, when {{{x > 2}}}, the curve is below the x axis.
Otherwise, the graph is either on the x axis or above it.


So it's whenever {{{x <= 1}}} or when {{{x = 2}}} is when {{{-x^3 + 5x^2 - 8x + 4 >=0}}} is the case.


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Algebraic Approach:


We'll use the rational root theorem.
Since the first term is either -1 or +1, this means we can look at the plus/minus factors of the last term (4) to generate the list of all possible rational roots. 
That list is:
-1, +1, -2, +2, -4, +4
Of course something like +1 is the same as simply 1. I put the plus there to have it pair with its minus counterpart.


From here, we try all of these possible roots one at a time into the expression given.


Let's say we tried x = -1
f(x) = -x^3 + 5x^2 - 8x + 4
f(-1) = -(-1)^3 + 5(-1)^2 - 8(-1) + 4
f(-1) = 18
The result is not zero, so x = -1 is not a root of f(x)


On the other hand, x = 1 is a root because it does lead to f(x) = 0
f(x) = -x^3 + 5x^2 - 8x + 4
f(1) = -(1)^3 + 5(1)^2 - 8(1) + 4
f(1) = 0


If you checked the others, you'd find that only x = 2 is the other root. In fact, it's a double root as the graph previously showed.
This means the cubic factors to -(x-1)(x-2)^2


Once you know the roots are x = 1 and x = 2, you'll set up a number line with those values on it.
Then pick something to the right of x = 1. Let's say we picked x = -1 since we already checked it and found that f(-1) = 18.
This shows that if x < 1, then f(x) > 0


Now pick something between x = 1 and x = 2. Why not x = 1.5
f(x) = -x^3 + 5x^2 - 8x + 4
f(1.5) = -(1.5)^3 + 5(1.5)^2 - 8(1.5) + 4
f(1.5) = -0.125
The result is negative to indicate f(x) is negative on the interval 1 < x < 2, and this matches with what the graph shows.


Lastly, plug in something to the right of x = 2
I'll pick x = 3
f(x) = -x^3 + 5x^2 - 8x + 4
f(3) = -(3)^3 + 5(3)^2 - 8(3) + 4
f(3) = -2
Therefore, f(x) < 0 when x > 2


Putting everything together, we have {{{-x^3 + 5x^2 - 8x + 4 >=0}}} only true when {{{x <= 1}}} or when {{{x = 2}}}
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