If P(x) is a polynomial of degree 4 such that P(-1)=P(1)=5 and P(-2)=P(0)=P(2)=2 then find the maximum value of P(x) - Maths - Relations and Functions

Question

If P(x) is a polynomial of degree 4 such that P(-1)=P(1)=5 and
P(-2)=P(0)=P(2)=2 then find the maximum value of P(x)

Ghanshyam Dhakar · Accepted Answer

Dear student,
let px = ax4+bx3+cx2+dx+ep0 = 2=0+0+0+0+ee=2use p2 = 2  2 =16a+8b+4c+2d+216a+8b+4c+2d=08a+4b+2c+d = 0  
1use p-2 = 2  2 =16a-8b+4c-2d+216a-8b+4c-2d=08a-4b+2c-d = 0 
2use p1 = 5 = a+b+c+d+2a+b+c+d=3  
3use p-1 = 5 = a-b+c-d+2a-b+c-d=3  
4eq1+eq28a-4b+2c-d +8a+4b+2c+d=8a+2c=04a+c=0c=-4aeq3+eq4a+b+c+d+a-b+c-d=62a+2c=6a+c=3use c=-4aa-4a=3a=-1then c=4eq3-1-b+4-d=3b=-duse eq18a+4b+2c+d = 0 -8+4b+8-b=0b=0 then d=0px = -x4+4x2+2differentiate p'x = -4x3+8x=0-4xx2-2=0x=0, x=±2p'x = -4x3+8xdifferentiatep''x = -12x2+8p''0 = 8>0p''±2 = -12×2+8=-16<0  so px have maxima at x=2,-2px = -x4+4x2+2p2 = -4+8+2=6 answer 
Regards