httpsgyazocomafc7e50c27d0d5815833ea926465f2fdSolutiona As we

https://gyazo.com/afc7e50c27d0d5815833ea926465f2fd

Solution

a)

As we see, this is a strictly decreasing function of x, as it is a geometric sequence of common ratio 1/2. Thus, its maximum value is at the start of the distribution,

mode = 1 [answer]

b)

f(x) = 12x^2 (1 - x)

Thus, optimizing f(x) by setting f\'(x) = 0,

f\'(x) = 12 (2x) (1-x) + 12x^2(-1) = 24x - 24x^2 - 12x^2 = 24x - 36x^2 = 0

x(24 - 72x) = 0

Thus,

x = 0 or 1/3

At x = 0, f(x) = 0

At x = 1/3, f(x) = 8/9

Thus, the mode is x = 1/3. [answer]

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c)

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

Thus, optimizing f(x) by setting f\'(x) = 0,

f\'(x) = (1/2) [(2x) e^-x + x^2 (-e^-x)] = 0

2x - x^2 = 0

x(2-x) = 0

x= 0 or 2

If x = 0, f(x) = 0
If x = 2, f(x) = 2 e^-2

Thus, the mode is x = 2. [answer]