Question

A tangent to the curve y=6x-x² cuts the x axis at point P. Find the coordinates of P. Help me calc the coordinates please​

Answer 1

We know that the line is tangent at the point (2,8)

The derivative of the function at x=2 is

$f'(x) = 6-2x \implies f'(2) = 6-4=2$

So, the tangent line passes through the point (2,8) and has slope 2. The equation is

$y-8 = 2(x-2) \iff y = 2x+4$

This line crosses the x axis where y=0:

$0 = 2x+4 \iff 2x = -4 \iff x = -2$