Suppose that f(x)= e^(-0.5x^2) (A) Find all critical values of f, compute their average.  (B) Use interval notation to indicate where f(x) is increasing. (C) Use interval notation to indicate...

Suppose that f(x)= e^(-0.5x^2)

(A) Find all critical values of f, compute their average.

 

(B) Use interval notation to indicate where f(x) is increasing.


(C) Use interval notation to indicate where f(x) is decreasing.

(D) Find the x-coordinates of all local maxima of f, compute their average.

(E) Find the x-coordinates of all local minima of f, compute their average.

(F) Find the x-coordinates of all inflection points of f, compute their average.

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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A) You need to solve the equation f'(x) = 0 to find if the function has critical points such that:

`f'(x) = e^(-0.5x^2)*(-0.5x^2)'`

`f'(x) = e^(-0.5x^2)*(-x)`

`f'(x) = -x/sqrt(e^(x^2))`

You need to solve the equation `f'(x) = 0 =gt -x = 0 =gt x = 0.`

Hence, the function has a critical value at x = 0.

B) Notice that the derivative f'(x)>0 for `x in (-oo,0),`  hence the function increases over interval `(-oo,0).`

C) Notice that the derivative`f'(x)lt0`  for `x in (0,oo), ` hence the function decreases over interval `(0,oo).`

D) The function reaches its maximum at x = 0.

E) The function has no minimum value.

F) You need to solve f''(x) = 0 to verify if the function has inflection points such that:

`f''(x) = (-sqrt(e^(x^2)) + (2x^2*e^(x^2))/(2sqrt(e^(x^2))))/(e^(x^2))`

`f''(x) = (-sqrt(e^(x^2)) + x^2*e^(x^2)sqrt(e^(x^2)))/(e^(x^2))`

`f''(x) = (sqrt(e^(x^2))(-1 + x^2*e^(x^2)))/(e^(x^2))`

`f''(x) = 0 =gt -1 + x^2*e^(x^2) = ` 0

`x^2*e^(x^2) = 1 =gt e^(x^2) = 1/(x^2)`

`x^2 = ln(1/x^2)`

This is a transcendental equation and you may use graphical method to solve it:

Notice that the intersections between the black curve and the red curve represents the inflection points of the graph of function, hence the x coordinates of inflection points are in interval (-1,1).

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