Consider the function f(x) = 5/4x^4 + 2/3x^3 - 5x^2 - 4x + 5
a. Find any relative extrema for f(x); be sure to label each as a maximum or minimum. You do not need to find function values; just find the x-values.
b. Determine the interval(s) where f(x) is increasing (if any) and the interval(s) where f(x) is decreasing (if any).
c. Determine the number of inflection points for this function and find their x-coordinate(s).
d. Determine whether f(x) is concave up or down at the following points: x = -2, x = -1, x = 0, and x = 2. Use this information and the information in parts A - C to sketch this function.
I really need help, and any help is greatly appreciated.
y = 5/4x^4 + 2/3x^3 - 5x^2 - 4x + 5
y' = 5x^3 + 2x^2 - 10x - 4 = (5x+2)(x^2-2)
y" = 15x^2 + 4x - 10
a. extrema where y'=-
max if y" < 0, min if y" > 0
b. increasing where y' > 0
c. inflection points where y" = 0
d. concave up if y" > 0
Now just plug in your numbers and functions.
To check:
https://www.wolframalpha.com/input/?i=%285%2F4%29x%5E4+%2B+%282%2F3%29x%5E3+-+5x%5E2+-+4x+%2B+5
To find the relative extrema of a function, follow these steps:
1. Find the derivative of the function f(x) to obtain f'(x), which represents the rate of change of the function.
2. Set f'(x) equal to zero and solve for x to find the critical points or potential extrema.
3. Determine the nature of each critical point using the second derivative test or by analyzing the sign changes in f'(x) around the critical points.
a. Let's start with finding the critical points or potential extrema:
1. Differentiate f(x) using the power rule and the sum/difference rule:
f'(x) = (5/4) * 4x^3 + (2/3) * 3x^2 - 2 * 5x - 4
f'(x) = 5x^3 + 2x^2 - 10x - 4
2. Set f'(x) equal to zero:
5x^3 + 2x^2 - 10x - 4 = 0
b. To determine the intervals where f(x) is increasing or decreasing, follow these steps:
1. Identify the critical points found in part a.
2. Choose a test point within each interval formed by the critical points.
3. Evaluate f'(x) using the test point to determine its sign.
- If f'(x) > 0, the interval is where f(x) is increasing.
- If f'(x) < 0, the interval is where f(x) is decreasing.
c. To find the number of inflection points and their x-coordinates, follow these steps:
1. Find the second derivative of f(x) by differentiating f'(x):
f''(x) = d/dx[f'(x)] = d/dx[5x^3 + 2x^2 - 10x - 4]
2. Set f''(x) equal to zero and solve for x to find the potential inflection points.
d. To determine whether f(x) is concave up or down at specific points, follow these steps:
1. Evaluate the value of f''(x) at each point provided.
Now, let's proceed with solving each part of the problem step by step.