Calculus tangent line

Question

For some constant c, the line $y=4x+c$ is tangent to the graph of $f(x)=x^2+2$, what is the value of $c$?

I don’t understand how to find the value of c. Because it’s a tangent line I understand they touch at one point. Probably a dumb question, I just don’t understand.

Answer 1

You must impose , if $z$ is the point common between $f$ and $y=4x+c$, that

$z^2+2=4z+c$

$f’(z)=4$

So

$z^2-4x+4=(z-2)^2=c+2$

and

$f’(z)=2z=4 \to z=2$

so

$(2-2)^2=0=c+2$

and $c=-2$

Answer 2

Hint: solve the equation $$4x+c=x^2+2$$ and set the discriminant equal to zero. you will get

$$x_{1,2}=2\pm \sqrt{2+c}$$ so $$c=-2$$

Answer 3

Say you have a tangent at point $T(a,b)$. Then $f'(a) = 4$ and $f(a)=b$ and $b=4a+c$. So we have

$\bullet \;\; 2a=4\implies a=2$

$\bullet \;\; a^2+2=b\implies b=6$

$\bullet \;\; c=-b+4a\implies c=-2$