Let p (x) be a real polynomial of degree 4 having extreme values `x=1 and x=2.if lim_(xto0) (p(x))/(x^2)=1`, then `p(4)` is equal to

Options

A. 0
B. 16
C. 32
D. 64

Answer

B

Explanation

Let `p(x) =ax^4+bx^3 +cx^2+dx+e`. Then,
`p'(x)=4ax^3+3bx^2+2cx+d`.
Now,
`lim_(x to 0) (p(x))/(x^2)=1`.
` rArr lim _( x to 0) (ax^2+bx +c+(d)/(x)+(e)/(x^2))=1`
` rArr c=1, d=e=0`
`therefore p'(x)=4 ax^3+3bx^2+2x`
It is given that `p(x)` has extreme values at `x=1 and x=2`
`therefore p'(1)=0 and p'(2)=0`
` rArr a=(1)/(4) and b= -1`
` therefore p(x)=(1)/(4)x^4-x^3+x^2`
`rArr p(4)=64-64+16=16`.