The tangent to the curve y = ex drawn at the point (c, ec) intersects the line joining the points (c

Question

The tangent to the curve y = ex drawn at the point (c, ec) intersects the line joining the points (c, - 1, ec-1) and (c + 1, ec+1)

Accepted Answer

Slope of the line joining the points (c - 1, e^c-1) and (c + 1, e^c+1) is equal to
⇒ tangent to the curve y = e^x will intersect the given line to the left of the line x = c.
Alternative
The equation of the tangent to the curve y = e^x at (c, e^c) is
y - e^c = e^c(x - c) ..... (1)
Equation of the line joining the given points is
y - e^c-1 = [x -(c - 1)] ..... (2)
Eliminating y from (1) and (2), we get
[x - (c - 1)] [2 - (e - e^-1)] = 2e^-1
or x - c = < 0 ⇒ x < c.
⇒ the line (1) and (2) meet on the left of the line x = c.

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