I'm not able to solve this problem: Find the equation of the tangent to the function $f(x) = \frac{5}{x} + 2x$ which passes through the point $P=(0,-4)$. Also give the coordinates of the tangent point $B$ and the angle of intersection between $t$ and $f$.
I have already taken the derivative of the function: $f'(x) = -\frac{5}{x^2} +2$ but when I plugged $P=(0,-4)$ it did not work because I got $f'(0) = -\frac{5}{0^2} +2$. Then I got error because I can't divide by zero.
Thank you for helping!!
Best Answer
The tangent line to the point $(a, 5/a+2a)$ has the slope $f’(a)=-5/a^2+2$. So the tangent line equation is
$$y= f’(a)(x-a)+f(a)$$
$$y= (-5/a^2+2)(x-a)+ 5/a+2a$$
$$y= (-5/a^2+2)x+10/a.$$
Put $(0,-4)$ in this: $$-4=10/a.$$
So $$a=-2.5.$$
The coordinates of the intersection point are $$(-2.5, -7).$$ The tangent line equation is then $$y=1.2x-4.$$