Find the formula for a quartic polynomial whose graph is symmetric about the y-axis and has local maxima at (-2,7) and (2,7) and a y-intercept of 5. Solution see since the polynomial is symmetric about y axis hence f(x)=f(-x) let f(x)= ax4+mx3+bx2+nx+c now since f(x)=f(-x)... m = n = 0 (comparing co-efficients) hence f(x)=ax4+bx2+c now for maxima we diffrentiate, we get 4ax3+2bx=0, put x=2 for maxima...we get 32a+4b=0 ....(1) now as y intercept is 5 , hence f(0) is 5 , => c=5 and (-2,7) lies on graph...hence by putting in main equation we get, 7=16a+4b+5 => 8a+2b=1 ....(2) from equation 1 and 2 we get a= -1/8 and b=1 hence the polynomial is y=f(x)= (-1/8)x4 + x2 + 5.