Thursday

April 17, 2014

April 17, 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.

**Related Questions**

AP Calculous - let f be the function defined by |x-1|+2 for X<1 f(x)= ax^2-Bx...

calculous - 3.Given the function f defined by f(x)=2x^3-3x^2-12x+20 a.Find the ...

For the function defined by...HELP - For the function defined by f(x)=3x^2+5x+2...

Calculous - let f be the function defined by f(x)=12x^2/3 -4X a)find the ...

math grade 11 - A quadratic function is defined by f(x)= 3x^2+4x-2. A linear ...

math grade 11 - A quadratic function is defined by f(x)= 3x^2+4x-2. A linear ...

Calculous - let f be the function defined by |x-1|+2 for X<1 f(x)= ax^2-Bx, ...

pre-calculus - for the function f defined by: (x)=3x^2+3x+2;find the following ...

Math (is this correct) - The functions f and g are defined by: f:x 3x+1 g:x 9-x ...

Calculus - A function f(x) is said to have a removable discontinuity at x=a if: ...