Given the quadratic function y=ax^2+bx+cy=ax2+bx+c, the maximum value is a^2+ 4a2+4, at x =1x=1, and the graph passes through point (3, 1). How do you find the values of the constants a, b, c?

1 Answer
Jan 12, 2017

(a,b,c)=(-3,6,10), or, (-1,2,4).(a,b,c)=(3,6,10),or,(1,2,4).

Explanation:

The graph of the function (fun.) y=ax^2+bx+cy=ax2+bx+c passes through

the point (pt.) (3,1)(3,1), so, the co-ordinates must satisfy the

equation.

:. 1=9a+3b+c...................(1)

y" is maximum at "x=1 :. [dy/dx]_(x=1)=0 :. 2a+b=0......(2)

Next,

y_(max)=a^2+4" occurs at "x=1 :. a+b+c=a^2+4.......(3)

(1),(2), &,(3) rArr (a,b,c)=(-3,6,10), or, (-1,2,4).

For y_(max)" at x=1, we must have, "[(d^2y)/dx^2]_(x=1) <0

dy/dx=2ax+b rArr (d^2y)/dx^2=2a=-6 or -2," readily "<0.

It is easy to verify that the pair of triads so derived satisfies the given

conditions.

"Therefore, "(a,b,c)=(-3,6,10), or, (-1,2,4).

Enjoy Maths.!