Thanks, that was pretty simple

Find two quadratic functions f and g such that$\displaystyle f (1) = 0, g(1) = 0$ and $\displaystyle f (0) = 10, g(0) = 10$ and both have a maximum value of 18.

So if $\displaystyle f(x) = ax^2 + bx + c$ then $\displaystyle c= 10$ for both f and g.

I also got $\displaystyle 0=a+b+10$ for both f and g

what do I do from here?

2. Originally Posted by jgv115
Thanks, that was pretty simple

Find two quadratic functions f and g such that$\displaystyle f (1) = 0, g(1) = 0$ and $\displaystyle f (0) = 10, g(0) = 10$ and both have a maximum value of 18.

So if $\displaystyle f(x) = ax^2 + bx + c$ then $\displaystyle c= 10$ for both f and g.

I also got $\displaystyle 0=a+b+10$ for both f and g

what do I do from here?
the question basically is that how many quadratic functions $\displaystyle f(x)=ax^2+bx+c$ are there satisfying
$\displaystyle f(1)=0, f(0)=10$ and the maximum value attained by f is 18.

since 1 is a root of $\displaystyle f$, we can write
$\displaystyle f(x)=k(x-1)(x-p)$; p being the other root.
given that $\displaystyle f(0)=10$, we have $\displaystyle kp=10$.
also maximum value of $\displaystyle f$ is 18 and we know that the maxima occurs at $\displaystyle x=(p+1)/2$ as p and 1 are the roots of f, we have
$\displaystyle f((p+1)/2)=18=(-k/4)((1-p)^2)$, which simplifies to,
$\displaystyle k((1-p)^2)+72=0$. now multiply both sides by k to get,
$\displaystyle (k-kp)^2+72k=0$. we already have the value of $\displaystyle kp$ which is 10, so we have,
$\displaystyle (k-10)^2+72k=0$ which gives two values of k which are $\displaystyle k=-2, k=-50$. both are acceptable as k had to be negative( if k is not negative then maxima of f does not exist).
corresponding values of p are -5 and -0.2.
so we have two such quadratic functions which can be found out by putting the values of k and p in $\displaystyle f(x)=k(x-1)(x-p)$

3. Originally Posted by jgv115
Thanks, that was pretty simple

Find two quadratic functions f and g such that$\displaystyle f (1) = 0, g(1) = 0$ and $\displaystyle f (0) = 10, g(0) = 10$ and both have a maximum value of 18.

So if $\displaystyle f(x) = ax^2 + bx + c$ then $\displaystyle c= 10$ for both f and g.

I also got $\displaystyle 0=a+b+10$ for both f and g

what do I do from here?
to proceed from here you have to exploit on the information that maxima of $\displaystyle f$ is 18.
you must already be aware that the maxima(or minima) occurs at $\displaystyle x=(r1+r2)/2$ where $\displaystyle r1$ and $\displaystyle r2$ are the roots of the quadratic equation.
now you must also be knowing that sum of roots can be expressed in terms of the coefficients of powers of x, viz, $\displaystyle r1+r2=(-b/a)$.
so you will get that maximum of f occurs at $\displaystyle x=(-b/2a)$.
so use $\displaystyle f(-b/2a)=18$. from this you will get one more relation in a and b. knowing that $\displaystyle a+b+10=0$ your problem can be solved.