Home
Class 12
MATHS
Show that the equation x^(3)+2x^(2)+x+5=...

Show that the equation `x^(3)+2x^(2)+x+5=0` has only one real root, such that `[alpha]=-3`, where `[x]` denotes the integral point of `x`

Text Solution

Verified by Experts

We have `x^(3)+2x^(2)+x+5=0`
`impliesx^(3)+2x^(2)+x=-5`
Let `f(x)=x^(3)+2x^(2)+x` and `g(x)=-5`
`:'f'(x)=0implies3x^(2)+4x+1=0`
`impliesx=-1,-1/3` and `f''(x)=6x+4`
`:.f'(-1)=-2lt0` and `f''(-1/3)=-2+4=2gt0`
`:.f(x)` local maximum at `x=-1` and local minimum at `x=-1/3`
and `f(-1)=0,f(-1/3)=-4/27`
Promotional Banner

Similar Questions

Explore conceptually related problems

Determine k so that the equation x^(2)-3kx+64=0 has no real roots.

Find roots equation 2x^(2)-x-3=0 .

Solve the equation 3x^(2)+ 5x+2=0 find the roots.

If the equation x^(3) +px +q =0 has three real roots then show that 4p^(3)+ 27q^(2) lt 0 .

The roots of the equation x^(2)-2sqrt(3)x+3=0 are

Integrate the function (x+3)/(x^(2)-2x-5)

How many real solutions of the equation 6x^(2)-77[x]+147=0 , where [x] is the integral part of x ?

If the equation x^(4)+px^(3)+qx^(2)+rx+5=0 has four positive real roots, find the maximum value of pr .

If 2a+3b+6c = 0, then show that the equation a x^2 + bx + c = 0 has atleast one real root between 0 to 1.

Find the roots of the equation 2x^(2)-5x+3=0 , by factorisation.