Find the set of values of k,where k is a constant,when equation x^3-12x^2+45x-34=k has (a) one root (b) three roots?

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Find the set of values of k,where k is a constant,when equation x^3-12x^2+45x-34=k has (a) one root (b) three roots?

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Answer 1 · 2018-03-28T20:04:53

$(a)$ $color(red)((a))$ , $20 < k < 16$ $20 < k < 16$

$(b)$ $color(red)((b))$ , $16 < k < 20$ $16 < k < 20$

Explanation:

.

$y = x^{3} - 12 x^{2} + 45 x - 34 = k$ $y=x^3-12x^2+45x-34=k$

We know that this is a cubic function and ordinarily would have three roots, i.e. it would cross the $x$ $x$ -axis in three points.

We also know that a cubic function typically has one maximum and one minimum unless the function is a perfect cubic in which case it would cross the $x$ $x$ -axis in one point which would be its inflection point without any maximum or minimum.

A perfect cubic function would be in the form of:

$y = {(x - a)}^{3}$ $y=(x-a)^3$

If we expand this we would get:

$y = x^{3} - 3 a x^{2} + 3 a^{2} x - a^{3}$ $y=x^3-3ax^2+3a^2x-a^3$

Setting the corresponding coefficients equal to each other, we get:

$- 3 a = - 12, 3 a^{2} = 45, and - a^{3} = - 34 - k$ $-3a=-12, 3a^2=45, and -a^3=-34-k$

This set of three equation does not have a solution as is evident. Therefore, our function can not be a perfect cubic. This means it has a minimum and a maximum and an inflection point.

$(a)$ $color(red)((a))$

In order for it to have one root, it would have to cross the $x$ $x$ -axis in one point only. This means that its minimum and maximum points would have to both be either above the $x$ $x$ -axis or below the $x$ $x$ -axis.

Let's find these points. We will take the derivative of the function and set it equal to $0$ $0$ :

$\frac{d y}{d x} = 3 x^{2} - 24 x + 45 = 3 (x^{2} - 8 x + 15)$ $dy/dx=3x^2-24x+45=3(x^2-8x+15)$

$x^{2} - 8 x + 15 = 0$ $x^2-8x+15=0$

$(x - 3) (x - 5) = 0$ $(x-3)(x-5)=0$

$x = 3 and 5$ $x=3 and 5$

Let's plug them into the function to find their $y$ $y$ values:

$x=3, :. y=20-k$

$x=5, :. y=16-k$

For these points to be above the $x$ -axis, their $y$ values have to be positive:

$y=20-k>0$ and $y=16-k>0, :. K<20 and k<16$

We have to go with $k<16$ to keep both points above the $x$ -axis. The following graph is drawn for $k=15$ :

enter image source here

For the points to be below the $x$ -axis, their $y$ values would have to be negative:

$y=20-k<0 and y=16-k<0, :. k>20 and k>16$

We have to go with $k>20$ to keep both points below the $x$ -axis. The following graph is drawn for $k=21$ :

enter image source here

Therefore, the condition for one root is:

$20 < k < 16$

$color(red)((b))$

For the function to have three roots, $k$ would have to be outside the range specified in $color(red)((a))$ .

Please note that if $k=16 or 20$ the function will have two roots because the graph will have either the maximum or the minimum touch the $x$ -axis and not cross it

The following graph is drawn for $k=16$ :

enter image source here

The following graph is drawn for $k=20$ :

enter image source here

Therefore,

$16 < k < 20$ for the function to have three roots. The following graph is drawn for $k=18$ :

enter image source here

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Find the set of values of k,where k is a constant,when equation x^3-12x^2+45x-34=k has (a) one root (b) three roots?

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Explanation:

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