AP Calculous
posted by Yoona .
let f be the function defined by f(x)=3X^5 5X^3 +2
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
f is increasing when f' is positive
f' = 15x^4  15x^2 = 15x^2 (x^21)
So, f' > 0 when x > 1
f is concave upward when f'' is positive
f'' = 60x^3  30x = 30x(2x1)
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
Oops. f'' = 30x(2x^2  1)
so 1/√2 < x < 0 or x > 1/√2 
AP Calculous 
drwls
a) That would be where the derivative
f'(x) = 15x^4 15x^2 > 0
x^2*(x^21) >0
Since x^2 must be positive or zero,
(x+1)(x1) > 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.
Respond to this Question
Similar Questions

Calculus
Let f be the function defined for x >or= to 0 with f(0)=5 and f', the first derivative of f, give by f'(x)=e^(x/4)sin(x^2). A) Use the graph of f' to determine whether the graph of f is concave up, concave down, or neither on the … 
Calculus  Functions?
#1. A cubic polynomial function f is defined by f(x) = 4x^3 +ax^2 + bx + k where a, b and k are constants. The function f has a local minimum at x = 1, and the graph of f has a point of inflection at x= 2 a.) Find the values of a … 
calculous
3.Given the function f defined by f(x)=2x^33x^212x+20 a.Find the zeros of f b.Write an equation of the line perpendicular to the graph of f at x = 0 c. Find the x and y coordinates of all points on the graph of f where the line tangent … 
Calculus c
Let f be a twicedifferentiable function defined on the interval 1.2 less than or equal to x less than or equal to 3.2 with f(1)=2. The graph of f', the derivative of f, is shown on the right. The graph of f' crosses the xaxis at … 
Calculus
Let f be a twicedifferentiable function defined on the interval 1.2 less than or equal to x less than or equal to 3.2 with f(1)=2. The graph of f', the derivative of f, is shown on the right. The graph of f' crosses the xaxis at … 
Calculus
1. On what interval is the function f(x)=x^34x^2+5x concave upward? 
Please check my Calculus
1. Given f(x)=6/x, choose the correct statement A. The graph of f is concave upward on the interval (negative infinity, 0) B. The graph of f is concave downward on the interval (negative infinity, 0) C. The graph of f is concave upward … 
Calculus
Consider the following function f(x)=x^2/[x^29] f(x) is increasing on the interval(s) f(x) is decreasing on the interval(s) f(x) has 2 vertical asymptotes x= f(x) is concave up on the interval(s) f(x) concave down on the interval(s) … 
Calculus(urgent please help)
Consider the following function f(x)=x^2/[x^29] f(x) is increasing on the interval(s) f(x) is decreasing on the interval(s) f(x) has 2 vertical asymptotes x= f(x) is concave up on the interval(s) f(x) concave down on the interval(s) … 
Calculus
The function g is defined for x>0 with g(1)=2, g'(x)=sin(x+1/x), and g"(x)=(11/x^2)cos(x+1/x). A. Find all values of x in the interval 0.12<=x<=1 at which the graph of g has a horizontal tangent line. B. On what subintervals …