Answer:
[tex]\displaystyle \\\left(\frac{49}{900},\frac{3761}{900}\right)[/tex] or approximately (0.544, 4.179)
Step-by-step explanation:
A function and its tangent lines intersect when their slopes are the same. Find the x-coordinate when the slope of f(x) is equal to 8/7 by taking the derivative of f(x):
[tex]\displaystyle\\f(x)=\sqrt{x}-x+4\\f'(x)=\frac{1}{2}\cdot\frac{1}{\sqrt{x}}-1=\frac{1}{2\sqrt{x}}-1[/tex]
Set [tex]f'(x)[/tex] equal to 8/7 and solve for x:
[tex]\displaystyle \\\frac{1}{2\sqrt{x}}-1=\frac{8}{7},\\x=\frac{49}{900}[/tex]
Therefore, [tex]f(x)[/tex] will intersect at a point of tangency with a line of slope 8/7 at x=49/900. Plug in x=49/900 into [tex]f(x)[/tex] to get the y-coordinate:
[tex]\displaystyle\\y=\sqrt{x}-x+4 \vert_{x=49/900}=\frac{3761}{900}[/tex]
⇒Answer: (49/900, 3761/900) or approximately (0.544, 4.179)
