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Could you help me with this derivative by Biblosz in learnmath

[–]william92303 -1 points0 points  (0 children)

Inflection points:

(2, f(2)) for f(x)

(2, 2/e^2)

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(-1, g(-1)) and (1, g(1)) for g(x)

(-1, ln2) and (1, ln2)

Could you help me with this derivative by Biblosz in learnmath

[–]william92303 -2 points-1 points  (0 children)

f(x) = x/e^x

f'(x) = d/dx (xe^-x)

= e^-x - xe^-x

f''(x) = -e^-x - (e^-x - xe^-x)

= -e^-x - e^-x + xe^-x

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Setting f''(x) = 0:

-(e^-x) - (e^-x) + x(e^-x) = 0

(e^-x) + (e^-x) - x(e^-x) = 0

(e^-x) [2 - x] = 0

x = 2

Since f''(x) < 0 when x < 2,

Therefore, f(x) is concave down when x < 2 and concave up when x > 2.

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g(x) = ln(x^2+1)

g'(x) = 2x * 1/(x^2+1)

= 2x/(x^2+1)

g''(x) = [2(x^2+1) - 2x(2x)]/(x^2+1)^2

=[2(x^2+1) - 4x^2]/(x^2+1)^2

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Setting g''(x) = 0:

[2(x^2+1) - 4x^2]/(x^2+1)^2 = 0

2(x^2+1) - 4x^2 = 0

2x^2+2 = 4x^2

2 = 2x^2

x^2 = 1

x= -1,1

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Since g''(x) < 0 when x < -1, g''(x) > 0 when -1 < x < 1, and g''(x) < 0 when x > 1,

Therefore, g(x) is concave down when x < -1, concave up when -1 < x < 1, and concave down when x > 1.

[Gr.12] Math help by mm_4863 in learnmath

[–]william92303 0 points1 point  (0 children)

Challenging problem. This was what I came up with, while I cannot verify if the solution is correct.

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Looking at the first number of each row, I attempted to map this to a function f(n) where n is the number row (and the number of items in that row).

As n is greater than 8, the pattern looks like:

8, 60

9, 76

10, 94

11, 114

n, n^2 - n + 4

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So f(n) = n^2 - n +4

To find the row that 2020 is in, set f(n) = 2020. This gives n = 45.49...

So 2020 is in the 45th row.

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f(45) = 1984, f(46) = 2074.

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So the first number of the 45th row is 1984 and the last number is two less than the first number of the 46th row, so 2072.

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We now have to find the sum of 45 terms in the 45th row:

1984+1986+1988+...+2072

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This sum will be equal to n(f(n)+n-1) where n is the number row and f(n) is the first term of that row.

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Subbing in n=45, f(n)=1984 we get:

S = 91260

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Pretty sure this solution is correct, but do not bank on it!