f(x) = 3x^3 - 9x + 5
find the:
1) zeroes or undefined values
2) intervals where the function is greater than zero
3) intervals where the function is less than zero
4) coordinates of all maxima and minima
5) intervals where the function is increasing
6) intervals where the function is decreasing
i'm not sure how to find this information using the equation. can someone help and explain? i'm allowed to use a graphic calculator.
graph it, and the graph will answer all those questions.
I am not certain how far you are in calculus. You can get them all analytically, also. Try your calculator first.
1) 1.3170051, 0.64500159, -1.962007
2) when x>1 and x= 0,-1
3) when x<-1 and x=-1
4) ???
5) x<1 x>1
6) x<1
is this correct? and i'm not sure how to find the maxima and minima
maxima and minima are where f' = 0
f = 3x^3 - 9x + 5
f' = 27x^2-9 = 9(3x^2-1)
so, when 3x^2-1 = 0, f has a max or a min.
To find the requested information about the function f(x) = 3x^3 - 9x + 5, you can utilize several methods, including graphing or algebraic techniques. Since you mentioned you have access to a graphic calculator, I will explain how to obtain the answers using both approaches.
1) To find the zeroes or undefined values, you need to determine the values of x where the function equals zero or becomes undefined. In this case, to find the zeroes, set f(x) = 0 and solve for x. To find the undefined values, look for any potential values of x that make the function undefined, such as dividing by zero.
Using Algebra:
To find the zeroes:
3x^3 - 9x + 5 = 0
Unfortunately, solving this cubic equation directly is not straightforward. However, you can use numerical methods like the Newton-Raphson method, which might be challenging to do by hand.
Using the Graphic Calculator:
Input the equation f(x) = 3x^3 - 9x + 5 into your graphic calculator and graph it. The x-intercepts (where the graph crosses the x-axis) represent the zeros of the function.
2) To determine the intervals where the function is greater than zero, you need to identify the x-values where the function is above the x-axis on the graph.
Using Algebra:
To find the intervals where f(x) > 0, you can examine the sign changes of the function at different x-values by factoring or using algebraic techniques such as the Rational Root Theorem.
Using the Graphic Calculator:
Input the equation f(x) = 3x^3 - 9x + 5 into your graphic calculator and observe the intervals on the x-axis where the graph is above zero.
3) To find the intervals where the function is less than zero, you need to identify the x-values where the function lies below the x-axis on the graph.
Using Algebra:
To find the intervals where f(x) < 0, you can examine the sign changes of the function at different x-values by factoring or using algebraic techniques.
Using the Graphic Calculator:
Input the equation f(x) = 3x^3 - 9x + 5 into your graphic calculator and observe the intervals on the x-axis where the graph is below zero.
4) To find the coordinates of all maxima and minima, you need to determine the highest and lowest points of the graph, called local extrema. These points have the highest or lowest y-values, respectively.
Using Algebra:
To find the coordinates of local extrema, you can calculate the derivative of the function and solve for x when f'(x) = 0. From there, find the corresponding y-values by substituting these x-values back into the original function.
Using the Graphic Calculator:
Input the equation f(x) = 3x^3 - 9x + 5 into your graphic calculator and observe the highest and lowest points of the graph. Alternatively, you can use the calculator's maximum and minimum functions to find the coordinates programmatically.
5) To find the intervals where the function is increasing, you need to identify the x-values where the graph displays a positive slope (i.e., going upwards).
Using Algebra:
To find the intervals where f(x) is increasing, you can examine the sign of the derivative of the function (i.e., f'(x)). When f'(x) > 0, the function is increasing.
Using the Graphic Calculator:
Input the equation f(x) = 3x^3 - 9x + 5 into your graphic calculator and observe the intervals on the x-axis where the graph goes upwards.
6) To find the intervals where the function is decreasing, you need to identify the x-values where the graph displays a negative slope (i.e., going downwards).
Using Algebra:
To find the intervals where f(x) is decreasing, you can examine the sign of the derivative of the function (i.e., f'(x)). When f'(x) < 0, the function is decreasing.
Using the Graphic Calculator:
Input the equation f(x) = 3x^3 - 9x + 5 into your graphic calculator and observe the intervals on the x-axis where the graph goes downwards.
Remember, while the algebraic techniques require more calculations, they offer precise and detailed results. On the other hand, using a graphic calculator provides a visual representation of the function, making it easier to understand and identify key features.