How do i find the formula of this function so it has inflection points and a local max at a certain x value?

Well to start, you should try to image what this function looks like. Why does it have a maximum and why does it have two inflection points? Here's a graph of the function where I let a=0 and b=1.

http://www.wolframalpha.com/input/?i=plot+e^%28-x^...

Now intuition should tell you how changing a and b changes the function. b, because it is a constant in front of the function changes the width of the curve. So a larger b means a wider peak and a smaller b means a thinner peak. a moves the function around on the x-axis. Right away you should be able to see that letting a = 0 sets our peak at 0. This shifting x by some value a will shift our peak to that point. So without any math you can say that if you set a to 32, suddenly the peak (i.e. where the local maximum is) is at x=32. But let's actually prove this mathematically.

To get maximums you just take the derivative with respect to x:

y' = [ -2 * (x - a) ] / b * e^(-(x-a)^2)

Now when you normally solve for peaks, you set y' = 0 and solve for x, but we already know x=32 so we can get the equation:

0 = [ -2 * (32 - a) ] / b * e^(-(32-a)^2

No let's take the second derivative to solve for get the inflection points:

y'' = (2a/b) * [ -2 * (x - a) ] * e^(-(x-a)^2) - (2/b) * e^(-(x-a)^2) - (2x/b) * [ -2 * (x - a) ] / b * e^(-(x-a)^2)

y'' = (2/b) * e^(-(x-a)^2) * [ -2(x - a)^2 - 1 ]

Now again you can set y" = 0 and x can be either 36 or 28. This allows for you to be able to solve for both a and b.

Hope that helps.

ok know the inflection ingredient is the ingredient the place the function shift from the +ive to -ive or the alternative . so iy's like the max/mini factors . and relating to the 2d answer you're suitable , u discover the mini and max from the 1st spinoff and u equivalent it to 0 and after that u take any ingredient on the best and the raise if it substitute into increasing and lowering then it extremely is max and the alternative it extremely is mini answer f '(x)= 3x^2 + x -2 =0 ==> (3x-2)(x+a million)=0 ==> x=2/3 and x=-a million so we now take any sort in the previous 2/3 and after 2/3 f '(0)= 3(0)^2 + 0 -2 = -2 so the -ive is mean that the unique function is lowering f '(a million)= 3(a million)^2 + a million -2 = 2 so the +ive is mean that the unique function is increasing so there is at x=2/3 close by mini fee for x=-a million f ' (-2) = 3(-2)^2 -2 -2 = 8 so the +ive is mean that the unique function is increasing f '(0)= 3(0)^2 + 0 -2 = -2 so the -ive is mean that the unique function is lowering so there is at x=-a million close by max fee