Home
Class 12
MATHS
Given P(x) =x^(4) +ax^(3) +bx^(2) +cx +d...

Given `P(x) =x^(4) +ax^(3) +bx^(2) +cx +d` such that `x=0` is the only real root of `P'(x) =0`. If `P(-1) lt P(1),` then in the interval `[-1,1]`

A

(a) P(-1) is the minimum and P(1) is the maximum of P

B

(b) P(-1) is not minimum but P(1) is the maximum of P

C

(c) P(-1) is not minimum and P(1) is not the maximum of P

D

(d) neither P(-1) is the minimum nor P(1) is the maximum of P

Text Solution

Verified by Experts

The correct Answer is:
2
`P(x) =x^(4)+ax^(3)+bx^(2)+cx+d`
`P(x)=4x^(3)+3ax^(2)+2bx+c`
Since x=0 is solution to P(x) =0 we get c=0 Therefore
`P(x)=x^(4)+ax^(3)+bx^(2)+d`
Also we have
`P(-1)ltP(1)`
`rarr 1-a+b+dlt1+a+b+drarragt0`
Since P(x) =0 only when x=0 and P(x) is differentiable in
(-1,1) we should have the maximum and minimum points x=-1,0 and 1 only
Also we have f(-1)`lt`P(1)
Maximum of P(x) =maximum {P(0),P(1)}and
Minimum of f(x)=minimum {P(-21),P(0)}
In the interval [0,1]
`P(x)=4x^(3)+3ax^(2)+2bx{(p(1)}` and
`4x^(2)+3ax+2b=0` has no real roots .Therefore
`(3a)^(2)-32blt0`
`rarr (9a^(2))/(32)ltb`
`rarrbgt0`
Thus we have `agt0 and bv gt0` Thus
P(x) =`4x^(3)+3ax^(2)+2bxgt0 forall x in (0,1)`
Hence P(x) is increasing in [0,1]
Similarly P(x) id P(1)
Therefore the minimum of P(x) does not occur at x=1
Doubtnut Promotions Banner Mobile Dark
|

Topper's Solved these Questions

  • MONOTONICITY AND MAXIMA MINIMA OF FUNCTIONS

    CENGAGE ENGLISH|Exercise Linked comprehension Type|2 Videos

  • MONOTONICITY AND MAXIMA MINIMA OF FUNCTIONS

    CENGAGE ENGLISH|Exercise Linked comprehension type|50 Videos

  • MONOTONICITY AND MAXIMA MINIMA OF FUNCTIONS

    CENGAGE ENGLISH|Exercise Numerical Value Type|24 Videos

  • METHODS OF DIFFERENTIATION

    CENGAGE ENGLISH|Exercise Single Correct Answer Type|46 Videos

  • MONOTONOCITY AND NAXINA-MINIMA OF FUNCTIONS

    CENGAGE ENGLISH|Exercise Comprehension Type|6 Videos

Similar Questions

Explore conceptually related problems

If the roots of ax^(3) + bx^2 + cx + d=0 are in G.P then the roots of dx^3 - cx^2 + bx -a=0 are in

Let P(x) =x^(2)+ 1/2x + b and Q(x) = x^(2) + cx + d be two polynomials with real coefficients such that P(x) Q(x) = Q(P(x)) for all real x. Find all the real roots of P(Q(x)) =0.

If P(x) = ax^(2) + bx + c , and Q(x) = -ax^(2) + dx + c, ax ne 0 then prove that P(x), Q(x) =0 has atleast two real roots.

Let P_n(x) =1+2x+3x^2+............+(n+1)x^n be a polynomial such that n is even.The number of real roots of P_n(x)=0 is

Let P_n(ix) =1+2x+3x^2+............+(n+1)x^n be a polynomial such that n is even.The number of real roots of P_n(x)=0 is

Let P_n(ix) =1+2x+3x^2+............+(n+1)x^n be a polynomial such that n is even.The number of real roots of P_n(x)=0 is

Let a, b, c in R with a gt 0 such that the equation ax^(2) + bcx + b^(3) + c^(3) - 4abc = 0 has non-real roots. If P(x) = ax^(2) + bx + c and Q(x) = ax^(2) + cx + b , then (a) P(x) gt 0 for all x in R and Q(x) lt 0 for all x in R . (b) P(x) lt 0 for all x in R and Q(x) gt 0 for all x in R . (c) neither P(x) gt 0 for all x in R nor Q(x) gt 0 for all x in R . (d) exactly one of P(x) or Q(x) is positive for all real x.

p: x = 5 or x = 1 is the root of equation (x - 5)^2 = 0, statement p is true since

In the given figure graph of y=p(x)=x^(4)+ax^(3)+bx^(2)+cx+d is given The product of all imaginary roots of p(x)=0 is (a) 1 (b) 2 (c) 1/3 (d) 1/4

Let f(x)= x^(3) + x + 1 and P(x) be a cubic polynomial such that P(0) = -1 and roots of f(0) = 1 ; P(x) = 0 are the squares of the roots of f(x) = 0 . Then find the value of P(4).

Home
Profile