FindA, B and C so that the lineAx+By=Cis tangent to the graph of f(x) =exand passes through (0,0).

Answer 1

The derivative of f(x) = e^x is f '(x) = e^x so the slope of the line is e^x. Using the point slope form of a line we get:

(y - 0) = e^x(x - 0)

y = xe^x

But the line must intersect f(x) = e^x so xe^x must equal e^x

xe^x = e^x

xe^x - e^x = 0

e^x(x - 1) = 0

so either e^x = 0 or (x - 1) = 0

e^x cannot equal zero, so we can through that out.

x - 1 = 0 when x = 1

So the slope is e¹ = e

And the equation of the line is then:

(y - 0) = e(x - 0)

y = ex

ex - y = 0

So A = e, B = -1, C = 0