Find the values of p for which the equation `x^(4)-14x^(2)+24x-3-p=0` has
(a) Two distinct negative real roots
(b) Two real roots of opposite sign
(c) Four distinct real roots
(d) No real roots

Explanation

To examine the roots of the equation `x^(4)-14x^(2)+24x-3-p=0` , let us first draw the graph of the function.
`f(x)=x^(4)-14x^(2)+24x-3`
`f'(x)=4x^(3)-28x+24`
`=4(x^(3)-7x+6)`
`=4(x^(3)-x^(2)+x^(2)-x-6x+6)`
`=4(x-1)(x^(2)+x-6)`
`=4(x-1)(x+3)(x-2)`
`f'(x)=0 :. x=1, 2, -3`
Also the leading coefficient is 1, when `x rarr +- oo , f(x) rarr oo`.
Sign scheme of `f'(x)` is as follows.

Clearly, from the sign scheme, x = -3 and x = 2 are the points of minima and x = 1 is the point of maxima.
`f(1)=8,f(2)=5` and `f(-3)=-120`
Also `f(0)=-3`
So the graph of `y=f(x)` is as shown in the following figure.

Now to examine the roots of the equation `x^(4)-14x^(2)+24x-3=p`, we draw the straight line y = p and find the intersection of `y=f(x)` and `y=p`.
Now for two distinct negative real roots of `f(x)=p, y=p` must cu t`y=f(x)` between point C and the x-axis. Hence the values of 'p' for two negative roots are (-120,0).
For two real roots of opposite sign, y = p must cut y = f(x) between points H and D. Hence the values of 'p' for two real roots of opposite sign are (-3,5).
For four distinct real roots, y=p must cut `y=f(x)` four times between points H and F. Hence the values of 'p' are (5,8).
For no real roots, `y=p` must lie below point C. Hence the values of 'p' are `(-oo,-120)`.