Find equations of the tangent lines to the curves y=f(x) andy=g(x) at x=a. Show that these tangent lines intersect on the x-axis at the point (a+ 1=c;0)

Question

dumbcow · Answer

equation of tangent line is given as:
$$y - f(a) = f'(a)(x-a)$$
$$\large f'(x) = -c e^{-cx}$$
$$\large y  = (-c e^{-ac})(x-a) + e^{-ac}$$
the same is done for g(x) in same way

then set the 2 tangent lines equal to find intersection point

dumbcow · Answer

$$\large \large (-c e^{-ac})(x-a) + e^{-ac}= (c e^{-ac})(x-a) - e^{-ac}$$
which simplifies to
$$\large 2c e^{-ac} x =2ace^{-ac}+ 2e^{-ac}$$
$$x = a + \frac{1}{c}$$