Determine the number of lines that are tangent to the curve y = ((e^(2x+1))*(x-2))/2 and pass through the point (0,-2).

Question

Determine the number of lines that are tangent to the curve y = ((e^(2x+1))*(x-2))/2 and pass through the point (0,-2).

Answer 1

Two.

Graphing it it is easy to see.

Analytically, (0,-2) is above (0,f(0)),,,-2>f(0)

the second derivative of f(x) is negative x=minus infinity to x=+1

meaning f(x) is concave down in that interval.

There is a point with x less than zero for a tangent line,

and there a point with x between 0 and +1 for a tangent line.

From first derivative, f(x) is decreasing x=minus infinity to x=+3/2

implying only one point each.

to find the points, must solve x²-2x-2e^-(2x+1)=0