Functions

[GMATSUCKER]

A quadratic function f(x) attains a maximum value of 3 at x=1. The value of the function at x=0 is 1.what is the value of f(x) at x =10 ?

a)-105

b)-119

c)-159

d)-110

e)-180

[sanju09]

A quadratic function f(x) attains a maximum value of 3 at x=1. The value of the function at x=0 is 1.what is the value of f(x) at x =10 ?

a)-105

b)-119

c)-159

d)-110

e)-180

Let f (x) = a x^2 + b x + c is the quadratic function, such that

At x = 1, a + b + c = 3, and at x = 0, c = 1

Then, at x = 10, f (x) = a (10) ^2 + b (10) + c = 100 a + 10 b + c = 90 a + 21 (?);

but f (x) cannot exceed 3, hence a is a negative integer.

Under these restrictions, no choice other than [spoiler]C[/spoiler] can take 90 a + 21, which happens at a = -2.

[spoiler]C[/spoiler]


I preferred to EXPLOIT the choices in hand, instead of going through the ways which are either long or beyond the GMAT specifications.

[GMATSUCKER]

Any other approach to solve this problem ?

[harshavardhanc]

Under these restrictions, no choice other than [spoiler]C[/spoiler] can take 90 a + 21, which happens at a = -2.

[spoiler]C[/spoiler]

I preferred to EXPLOIT the choices in hand, instead of going through the ways which are either long or beyond the GMAT specifications.

nice approach!

Any other approach to solve this problem ?

as said by sanju, the other methods to solve this problem will probably employ techniques that are beyond GMAT.

For e.g.

generic quad eq f(x) = ax^2 + bx + c (observe that there are 3 variables)

now based on the question, you can easily get value of C =1 and another eq. which gives a+b=2.

what about the third eq ?

you will have to differentiate the quadratic eq. with respect to x ( a concept beyond the scope), which will give you :

maximum value of this quad eq. is at x= (-b/2a). which on further solving will give a=-2 and b=4.

Hence the quad function is y= -2x^2 + 4x +1

at x=10, the value will be -159.

But, as a point to remember you can keep this in mind :

maximum/minimum value of a 2nd degree equation (quad function) in x represented by ax^2 + bx + c is at x= (-b)/2a

would like to see any simpler methods.