Let f be the function given by f(x) = tan x and let g be the function given by g(x) = x^2. At what value of x in the interval 0≤x≤π do the graphs of f and g have parallel tangent lines?

Answer 1

The slope of the tangent line of a function f(x) is given by the derivative of the function, f'(x).

Location of parallel tangent lines can be obtained by equating f'(x) and g'(x), and solving for x on the interval [0,π].
f(x)= tan(x), f'(x)=sec²(x)
g(x)= x², g'(x)=2x

Therefore, solve for x in
sec²(x) = 2x in 0≤x≤π

You can solve this graphically.
I get x=2.1 approx. by plotting.

You can also refine the solution using Newton's method:
f'(x)-g'(x)=0, so
sec²(x)-2x=0
x1=x0-(sec²(x)-2x)/(2sec²(x)tan(x)-2)
By iterating two or three times, I get the solution x=2.08284