Find the points of inflection and discuss the concavity of the graph of the...

[good_guy]

...function ? 1) f(x) = 2x^3 - 3x^2 - 12x + 5

2) f(x) = x.(rt. (x+1))

[Morgan]

Problem #1
-------------

Given:
f(x) = 2x^3 - 3x^2 - 12x + 5

Determine first derivative:
f'(x) = 6x^2 - 6x - 12

Determine second derivative:
f''(x) = 12x - 6

Find the zeros:
0 = 12x - 6
6 = 12x
1/2 = x

Solving for y:
y = 2x^3 - 3x^2 - 12x + 5
y = 2*(1/2)^3 - 3*(1/2)^2 - 12*(1/2) + 5
y = 2*(1/8) - 3*(1/4) - 12*(1/2) + 5
y = 1/4 - 3/4 - 6 + 5
y = - 1/2 - 1
y = - 3/2

Thus, the point of inflection is at: (1/2, -3/2).

To understand the concavity, test points to each side
of the point of inflection. To the left, try x=0:

f''(0) = 12*0 - 6 = -6
Thus, to the left of x=1/2, the second derivative is
negative which means the curve is concave downward.

f''(100) is clearly a large positive number. Thus,
the curve is concave upward to the right of x=1/2.

To verify this, look at the graph at:
http://www06.wolframalpha.com/input/?i=2x%5E3+-+3x%5E2+-+12x+%2B+5



Problem #2
-------------

f(x) = x.(rt. (x+1))

Is this an artifact of using some special font? Maybe ".(rt."
is supposed to mean "root", as in "square root"? Working with
that assumption, I'll guess that you meant:
f(x) = x * sqrt(x+1)

This can be rewritten as:
f(x) = x * (x+1)^(1/2)

Take the first derivative:
f'(x) = x * (1/2)(x+1)^(-1/2) + (x+1)^(1/2)
f'(x) = (1/2)x * (x+1)^(-1/2) + (x+1)^(1/2)


Take the second derivative:
f''(x) = (1/2)x * (-1/2)(x+1)^(-3/2) + (x+1)^(-1/2)*(1/2) + (1/2)(x+1)^(-1/2)
f''(x) = (-1/4)x * (x+1)^(-3/2) + (1/2)*(x+1)^(-1/2) + (1/2)(x+1)^(-1/2)
f''(x) = (-1/4)x * (x+1)^(-3/2) + (x+1)^(-1/2)
f''(x) = ( (-1/4)x / (x+1)^(3/2) ) + 1 / (x+1)^(1/2)
f''(x) = ( (-1/4)x / (x+1)^(3/2) ) + (x+1) / (x+1)^(3/2)
f''(x) = ( (-1/4)x + (x+1) ) / (x+1)^(3/2)
f''(x) = ( (3/4)x + 1 ) / (x+1)^(3/2)

Find the zeros:
0 = ( (3/4)x + 1 ) / (x+1)^(3/2)
0 = (3/4)x + 1
-1 = (3/4)x
-4 = 3x
-4/3 = x

However, this is not an inflection point in the Real number
solutions because the function is only defined in the domain
of -1 <= x < infinity.

If you evaluate the second derivative for any value of x greater than -1,
it is positive, which says the curve is concave upward (with no obvious
point of inflection).

To visualize this, see the graphs at:
http://www09.wolframalpha.com/input/?i=y+%3D+x+*+sqrt%28x%2B1%29




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