MHB Find a real number for a continuous function

[Maxers99]

How would I go about doing this?

Find a real number f so that: is a continuous function

y = { 3x - 2f if x is less than or equal to 0. }
{ 2x² + x + 5f² if x is less than 0 }

 

[anemone]

Hi Maxers99, welcome to MHB!:)

I am pretty sure you mean the domain for the second function is set for $x>0$, i.e. what we have here is the following piece-wise function:

$\displaystyle y(x)=\begin{cases}3x-2f & x\le0\\2x^2+x+5f^2 & x>0\\ \end{cases}$

Since $y$ is a continuous function, so at $x=0$, the two expressions must be equal. Can you proceed with this little hint?

 

[MarkFL]

At the risk of being somewhat redundant, to put what anemone has stated in the parlance of limits, we require:

$$\lim_{x\to0^{-}}(3x-2f)=\lim_{x\to0^{+}}\left(2x^2+x+5f^2 \right)$$

 

[Maxers99]

So if x=0, it would then be -2f = 5f^2

 

[anemone]

So if x=0, it would then be -2f = 5f^2

Correct!:)

But bear in mind that we are asked to find the real values of $f$. So now we have the quadratic equation in terms of $f$, i.e.

$-2f = 5f^2$

or

$ 5f^2+2f=0$

do you know how to solve this quadratic equation for $f$?

 

[HallsofIvy]

Correct!:)

But bear in mind that we are asked to find the real values of $f$. So now we have the quadratic equation in terms of $f$, i.e.

$-2f = 5f^2$

At this point, either f= 0 or we can divide both sides by f.

or

$ 5f^2+2f=0$

do you know how to solve this quadratic equation for $f$?