Paul S. answered • 11/19/14

Tutor

5
(11)
Patient & compassionate Math and Physics tutor

This is a Gaussian function, i.e. a bell curve. First, you should plot this in a computer or a graphing calculator.

To identify the critical points, take the derivative and set it to zero!!

g'(x) = e^(-6x^2) * (-6*2x) = -12x*e^(-6x^2)

g'(x) = 0 at only one point, i.e. x=0

Therefore, the function is flat (slope = 0) at x=0

The value of g(0) is 1, and g(x) < 1 for all x =/= 1 (again, check on a graphing calculator). Therefore, 0 is a global maximum.

Inflection points require the second derivative!!! Use the product rule in this case.

g''(x) = ( [-12*e^(-6x^2)]+[-12x*(-12x)*e^(-6x^2)] ) = (-12+144*x^2)*e^(-6x^2)

g''(x) = 0 at -12+144*x^2 = 0 ==> x = +/- Sqrt(1/12) = +/- 0.289

Therefore, the curvature of g(x) changes at x = Sqrt(1/12), on both sides of the y-axis (positive AND negative).

The value of g(+/- Sqrt(1/12)) is e^(-1/2) = 0.607

The slope of g(+/- Sqrt(1/12)) is g'(+/- Sqrt(1/12)) = -12*(+/-Sqrt(1/12))*e^(-1/2) = -/+ Sqrt(12)*e^(-1/2) = -/+ 2.101

Notice the slope at the inflection points is NEGATIVE on the POSITIVE SIDE of the y-axis, and POSITIVE on the NEGATIVE SIDE of the y-axis.

Hope this helps.

Best,

Paul