Given a function
f(x)2/3x^3+5/2x^2-3x.
a) Find
i. The inflection point.
ii. The y-intercept and x-intercept.
b) Sketch the graph of f(x).
i have already try it..but i don't understand.. which graph that is true.. the first or second ? and how to calculate the x,y intercept,and inflection point? i hope you can help me to solve this ..
y = 2/3 x^3 + 5/2 x^2 - 3x
y' = 2x^2 + 5x - 3
y'' = 4x + 5
inflection where y''=0 -- at x=-5/4
intercepts at (0,0)
and at x = 3/8 (-5±√57)
Looks like you need to review the meanings of 1st and second derivatives.
remember that the first derivative gives you the max's and min's and the second derivative tells you concavity and inflection points
To find the inflection point, y-intercept, and x-intercept, as well as sketch the graph of the function f(x) = (2/3)x^3 + (5/2)x^2 - 3x, let's break down the process step by step:
a) Inflection Point:
To find the inflection point, we need to calculate the second derivative and set it equal to zero.
1. Find the first derivative:
f'(x) = d/dx[(2/3)x^3 + (5/2)x^2 - 3x]
= (2/3)(3x^2) + (5/2)(2x) - 3
= 2x^2 + 5x - 3
2. Find the second derivative:
f''(x) = d/dx[2x^2 + 5x - 3]
= 4x + 5
3. Set the second derivative equal to zero and solve for x:
4x + 5 = 0
4x = -5
x = -5/4
The value of x = -5/4 is the x-coordinate of the inflection point.
b) Y-Intercept and X-Intercepts:
To find the y-intercept, substitute x = 0 into the function.
1. Y-Intercept:
f(0) = (2/3)(0)^3 + (5/2)(0)^2 - 3(0)
= 0 + 0 - 0
= 0
Hence, the y-intercept is (0,0).
2. X-Intercept:
To find the x-intercepts, set f(x) = 0 and solve for x.
(2/3)x^3 + (5/2)x^2 - 3x = 0
This equation might be challenging to solve analytically, so you can use numerical methods or a graphing calculator to approximate the x-intercepts. These methods involve finding the values of x for which f(x) = 0.
c) Sketching the Graph:
To sketch the graph of f(x), you can use the calculated inflection point, y-intercept, and x-intercepts.
1. Plot the inflection point (x, y):
The inflection point is at (x, f(x)), where x = -5/4. So, find the corresponding y-value by substituting x = -5/4 into the function f(x):
f(-5/4) = (2/3)(-5/4)^3 + (5/2)(-5/4)^2 - 3(-5/4)
After calculating, you will determine the value of f(-5/4) which will be the y-coordinate of the inflection point.
2. Plot the y-intercept (0,0).
3. Plot the x-intercepts approximately using the values obtained in step b).
4. Connect the plotted points to form the curve of the graph.
Regarding the two possible graphs you mentioned, without more information it is difficult to determine which one is correct. The graph will feature the inflection point, y-intercept, and x-intercepts we calculated earlier.
If you need a more precise graph, consider using a graphing calculator or graphing software to plot the function.
I hope this explanation helps you to solve the problem and sketch the graph correctly.