Question

the maximum point on curve x= e....?​

Answer 1

Equation of curve => x = y.e^x

y = x/e^x .............(a)

For maxima or minima:

dy/dx = 0

(x/e^x)' = 0

[e^x - x.e^x]/e^2x = 0

e^x(1 - x)/e^2x = 0

(1 - x)/e^x = 0

=> (1 - x) = 0

=> x = 1

Now, the point x = 1 can either be a point of maxima or it can be a point of minima.

So, for checking whether x = 1 is a point of maxima or minima we must study the sign of its first derivative at the limit x-»1.

Now,

lim(x-»1+) | (1 - x)/e^x > 0 => positive

and,

lim(x-»1-) | (1 - x)/e^x < 0 => negative

The derivative of y at lim(x-»1) changes its sign from positive to negative.

Hence, x = 1 is a point of Maxima.

Now, for the co-ordinate y at x = 1, put x = 1 in (a):

=> 1 = y.e^1

=> y = 1/e

=> y = e^-1

Hence, the maximum point on the curve x=y.e^x is (1, e^-1).

Therefore, the correct answer is:

✔️(b)-» (1, e^-1)