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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#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#-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#-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#

If we expand this we would get:

#y=x^3-3ax^2+3a^2x-a^3#

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

#-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.

#color(red)((a))#

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

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

#dy/dx=3x^2-24x+45=3(x^2-8x+15)#

#x^2-8x+15=0#

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

#x=3 and 5#

Let's plug them into the function to find their #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#:

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#:

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#:

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

Therefore,

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