Let f(x)= x^5-5x+3

1-Find f'(x) and f"(x).
2-Find the intervals on which f(x) is increasing or decreasing.
3-Find the local maximum and minimum of f(x), if any.
4-Find the intervals on which the graph of f(x) is concave up or concave down.
5-Find the points of inflection, if any.

To find the derivative, f'(x), of the given function f(x), use the power rule for differentiation. The power rule states that the derivative of x^n, where n is a constant, is n*x^(n-1). Applying this rule to each term in f(x), we have:

f(x) = x^5 - 5x + 3

1- Finding f'(x):
Applying the power rule:

f'(x) = 5*x^(5-1) - 5*x^(1-1) + 0

Simplifying the exponents and constants:

f'(x) = 5x^4 - 5

2- Finding the intervals of increasing and decreasing:
To determine when f(x) is increasing or decreasing, we need to examine the values of f'(x).

To find the critical points, set f'(x) equal to zero and solve for x:

0 = 5x^4 - 5

Adding 5 to both sides:

5 = 5x^4

Dividing both sides by 5:

1 = x^4

Taking the fourth root of both sides:

x = 1

Now, we need to test intervals on either side of x = 1 to check the sign of f'(x) in those intervals:

For x < 1, choose x = 0:
f'(0) = 5(0)^4 - 5 = -5

For x > 1, choose x = 2:
f'(2) = 5(2)^4 - 5 = 75

Based on the sign of f'(x) in each interval, we can conclude:
- When x < 1, f(x) is decreasing.
- When x > 1, f(x) is increasing.

3- Finding local maximum and minimum:
To find the local maximum and minimum of f(x), we need to locate any critical points and check the concavity of the function.

In this case, we have only one critical point at x = 1. To determine if it is a local maximum or minimum, we can use the second derivative test. If f''(x) is positive at the critical point, it indicates a local minimum. If f''(x) is negative, it indicates a local maximum.

Finding f''(x):
Applying the power rule:

f''(x) = 4 * 5x^(4-1) - 0

Simplifying:

f''(x) = 20x^3

Now, substitute x = 1 into f''(x):

f''(1) = 20(1)^3 = 20

Since f''(1) is positive (greater than zero), x = 1 is a local minimum point for f(x).

4- Finding the intervals of concavity:
To find the intervals of concavity, we need to determine when the second derivative, f''(x), is positive or negative.

Since f''(x) = 20x^3 is always positive (since the cube of any real number is positive), we can conclude that the graph of f(x) is concave up for all values of x.

5- Finding the points of inflection:
To find the points of inflection, we need to locate any values of x where the concavity changes. In this case, since the graph of f(x) is concave up for all x, there are no points of inflection.

To summarize:
1- f'(x) = 5x^4 - 5
2- f(x) is decreasing for x < 1 and increasing for x > 1.
3- The local minimum of f(x) is at x = 1.
4- The graph of f(x) is concave up for all x.
5- There are no points of inflection for f(x).