Question

Question #aa795

Answer 1

`x=+-pi/2`. But these are outside the open interval #(-pi/2, pi/2)). So, within here, there is no point of inflexion.

Explanation:

Let `y = e^x sin x`.

`y'=(s^x)'sin x+e^x(sin x)'`

`=e^x(sin x + e^x cos x`

`= e^x(sin x + cos x )`

`y''=(e^x)'(sin x + cos x )+e^x(sin x +cos x)'`

`=e^x(sin x + cos x + cos x - sin x)`

`=2e^x cos x=0, when cos x =0 to x=+-pi/2`

`y'''=2((e^x)'cos x+e^x(sin x)'`

`=2e^x(cos x-sin x)`

At `x=+-pi/2`, y'''is not 0##

Thus,`x = +-pi/2` are points of inflection. But these ar outside `(-pi/2, pi/2)`.

Note that y'=0 is not a necessary condition.

Here, y' is not 0 at `x = +-pi/2`.