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August 30, 2014

Posted by **Yoona** on Sunday, November 20, 2011 at 2:33pm.

a) on what interval is f increasing? b) on what interval is the graph of f concave upward?

c)Write the equation of each horizontal line tangent to the graph of f

- AP Calculus -
**Steve**, Sunday, November 20, 2011 at 2:48pmf is increasing when f' is positive

f' = 15x^4 - 15x^2 = 15x^2 (x^2-1)

So, f' > 0 when |x| > 1

f is concave upward when f'' is positive

f'' = 60x^3 - 30x = 30x(2x-1)

So, f'' > 0 when x < 0 or x > 1/2

Horizontal lines have slope=0. So, we want places where f'(x) = 0

15x^2 (x^2 - 1) = 0

x = -1, 0, 1

The horizontal lines are

y=f(-1)

y=f(0)

y=f(1)

evaluate f(x) at those points to get your lines.

- AP Calculous -
**Steve**, Sunday, November 20, 2011 at 2:51pmOops. f'' = 30x(2x^2 - 1)

so -1/√2 < x < 0 or x > 1/√2

- AP Calculous -
**drwls**, Sunday, November 20, 2011 at 2:52pma) That would be where the derivative

f'(x) = 15x^4 -15x^2 > 0

x^2*(x^2-1) >0

Since x^2 must be positive or zero,

(x+1)(x-1) > 0

x > 1 or x<-1

b) That would be where f"(x) > 0

c) Horizontal tangents would be where f'(x) = 0.

Find those x and y values.

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