Identify the x-intercepts, local maximum, and local minimum of the graph of
f(x)=x^3+2x^2-13x+10
I am confused on where to begin.
Also my other question asked to perform the indicated operation AND STATE THE DOMAIN.
(5x+7)*(x-9) = 5x^2 -38x-63 < ANSWER
How would I write my Domain?
I also wasn't sure how to put the equation together..
perform the indicated operation...
g(f(x)); f(x)=x^2+3x; g(x)=2x+1
-5, 1 , 2
http://www.wolframalpha.com/widgets/view.jsp?id=a7d8ae4569120b5bec12e7b6e9648b86
(right away you should see that one works, then you have a quadratic)
domain of parabola is all real x. (range is another story)
g = (2x+1)^2 + 3(2x+1)
now you multiply out and collect like terms
On the first one, it's easy enough to find the intercepts. Coming up with the max and min is a bit tricker, if you don't have calculus as a tool.
x = -b/2a which gives you a turning point which will help you come up with the max or the min.
Once you fine x, then you can find y by substituting back into the original equation.
Range can also be found by using this turning point.
x = -b/2a only works with quadratic functions. When the polynomial is of degree 3 or higher, all bets are off.
In this case, there is a local max or min when 3x^2+4x-13=0
To identify the x-intercepts, local maximum, and local minimum of the graph of a function, you will need to analyze its first and second derivatives. Here are the steps to find them:
1. Start by finding the first derivative of the function f(x).
For the given function f(x) = x^3 + 2x^2 - 13x + 10, differentiate it with respect to x:
f'(x) = 3x^2 + 4x - 13
2. Set f'(x) equal to 0 and solve for x to find critical points. These are the points where the slope of the function is either 0 or undefined, potentially indicating a local maximum or minimum.
Setting f'(x) = 0 and solving for x:
3x^2 + 4x - 13 = 0
Using factoring, completing the square, or the quadratic formula, you can find the values of x that make the above equation true.
3. Once you have found the critical points, plug them into the second derivative of f(x) to determine whether they correspond to local maximum or minimum points.
Differentiate f'(x) = 3x^2 + 4x - 13 to find the second derivative f''(x).
f''(x) = 6x + 4
Evaluate f''(x) at each of the critical points found in step 2. If f''(x) > 0 at a particular critical point, it corresponds to a local minimum, and if f''(x) < 0, it corresponds to a local maximum.
4. Finally, to find the x-intercepts, set f(x) = 0 and solve the equation for x.
For the given function f(x) = x^3 + 2x^2 - 13x + 10, set f(x) = 0:
x^3 + 2x^2 - 13x + 10 = 0
Use factoring, long division, synthetic division, or numerical approximation techniques (e.g., graphing calculator, numerical solver, etc.) to find the x-values that make the equation true.
By following these steps, you should be able to identify the x-intercepts, local maximum, and local minimum of the given function.