An algebra problem by A Former Brilliant Member

Algebra Level 4

Find the largest natural number $a$ for which the maximum value of $f(x) = a - 1 + 2x - x^{2}$ is smaller than minimum value of $g(x) = x^{2} -2ax +10 -2a$

The answer is 1.

1 solution

Guilherme Niedu
Oct 27, 2016

Differentiating, the minimum value for $f(x)$ is $a$ and the maximum value for $g(x)$ is $10 - 2a - a^2$ . So:

$a < 10 - 2a - a^2$

$a² + 3a - 10 < 0$

Which leads to:

$a \in (-5, 2)$

The largest natural number within the specified interval is $\fbox{1}$

We can do this without calculus by using the fact that maximum or minimum of a quadratic equation is -D/4a where "D" is the discriminant and "a" is the coefficient of x^2 .If a > 0 then minimum exists and if a < 0 maximum exist.

Aaron Jerry Ninan - 4 years, 7 months ago

Yup but doing differentiation is also very easy here!

Prakhar Bindal - 4 years, 7 months ago

But,i did it in other way and got the answer as 2.

The max value f(x) is when x= 1

Therefore f(x)=a

g(x)=11-4a

f(x)<g(x)

a<11-4a

a<11/5

Therefore a must be 2

genis dude - 4 years, 7 months ago

Sorry I got 1 I shouldn't have substituted x= 1 g(x)

genis dude - 4 years, 7 months ago

An algebra problem by A Former Brilliant Member

The answer is 1.

1 solution

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