Given the quadratic function #y=ax^2+bx+c#, the maximum value is #a^2+ 4#, at #x =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

Answer:

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

Explanation:

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

the point (pt.) #(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.!