The value 5 is a an upper bound for the zeros of the function shown below f(x)=x^4+x^3-11x^2-9x+18
1 answer:
Answer:
Answer to this question is TRUE.
Step-by-step explanation:
The graph of the function:
is attached to the answer.
Clearly by looking at the graph we could see that the zeros of the function f(x) are <em>-3, -2, 1 and 3.</em>
All the zeros are distinct.
<em>" Also upper bound of a number 'p' means the set of all those numbers which are greater than 'p' ".</em>
Here 5 will be an upper bound for the zeros of the function f(x) since all the 4 zeros of f(x) are less than 5.
Hence, the given statement that the value 5 is an upper bound for the zeros of the function f(x) is TRUE.
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I let the computer do the graphing.
f(x) = x(x + 3)²(1 - x) / 3
We see obvious zeros at x=0, x=-3 and x=1.
f(x) = x(x + 3)²(1 - x) / 3
f(x) = x(x^2 + 6x + 9)(1 - x)/3
f(x) = x(x^2 + 6x + 9 - x^3 - 6x^2 - 9x)/3
f(x) = x(- x^3 - 5x^2 -3x +9)/3
f(x) = (-x^4 - 5x^3 -3x^2 +9x)/3
f'(x) = (-4x^3 - 15x^2 -6x + 9)/3
We know there's f'(-3)=0 so x+3 must be a factor here,
f'(x) = (x + 3) (-4x^2 - 3x + 3)
We get stationary points when x = -3 or
x = (-1/8)( 3 ± √(9 - 4(-4)(3))) = (-3 ± √57)/8 about -1.3 and 0.6 which are a local minimum and maximum respectively.
f''(x) = (-12x^2 - 30x -6)/3 = -4x^2 - 10x - 2
f''(-3) = -4(9)+ 30 - 2 = -8
Concave Down, Negative. (-3,0) is a local maximum.
f''((-3 - √57)/8) = -4(-1.3)^2 - 10(-1.3) - 2 = 4.24
Concave up, positive, a CUP
(-3 - √57)/8, f((-3 - √57)/8) ) is a minimum.
f''((-3 + √57)/8) = -4(.6)^2 - 10(.6) - 2 = -9.44
CDN, ( ((-3 + √57)/8), f((-3 + √57)/8) ) is a maximum.
I didn't evaluate f((-3 ±√57)/8) which are around -3 and 1 but I leave that calculation to you.
Answer:
Part 1) The y-intercept is the point (0,-3)
Part 2) The slope in fraction form is
Part 3) The graph in the attached figure
Step-by-step explanation:
we have
Part 1) What is the y-intercept?
Remember that
The y-intercept is the value of y when the value of x is equal to zero
so
For x=0
substitute in the linear equation and solve for y
The y-intercept is the point (0,-3)
Part 2) What is the slope (in fraction form)?
we have
This is the equation of the line in slope intercept form
where
m is the slope or unit rate
b is the y-intercept
so
In fraction form is
The slope is negative, that means, is a decreasing function (as the x-values increases the y-values decreases)
Part 3) Graph the line
To graph the line we need two points
From the y-intercept (0,-3) use the slope to find your next point
The slope is -2, so you will rise -2 (2 units down) and run 1 (1 unit to the right).
The next point is (0+1,-3-2) -----> (1,-5)
so
Plot the points (0,-3) and (1,-5), connect them and graph the line
The graph in the attached figure
