Question

what is a first derivative test and second derivative test?

show some examples...!

Answer 1

I believe the 1st derivative test will tell whether the function is increasing or decreasing at a certain point or interval.
If the derivative or slope is positive, then function is increasing. If negative, its decreasing.
Ex:
\[f(x) = 5x^{3}-3x+1\]
\[f'(x) = 15x^{2}-3\]
test at point x=1
\[f'(1) = 15(1) -3 = 12\]
12 > 0 --> f(x) is increasing at x=1
The 2nd derivative test will determine the concavity of the function at a point or interval.
If 2nd derivative is positive, function is concave up. Negative, function is concave down.
Using the same example, lets take the 2nd derivative:
\[f''(x) = 30x\]
test at x=1 again:
\[f''(1) = 30(1) = 30\]
30 > 0 --> f(x) is concave up at x=1

Lets evaluate other points, x=0 and x= -1
\[f'(-1) = 15(-1)^{2} -3 = 12\]
\[f''(-1) = 30(-1) = -30\]
At x=-1 , f(x) is increasing but concave down.
\[f'(0) = 15(0)-3 = -3\]
\[f''(0) = 30(0) = 0\]
At x=0, f(x) is decreasing and changing concavity. Going from concave down to concave up.
In the interval [-1,1] we see f(x) transition from increasing to decreasing and back to increasing, also during the interval f(x) changes concavity from concave down to concave up.
This is how we can use 1st and 2nd derivative tests to evaluate a functions behavior for a given interval or at a specific point.
It also can help in graphing a function by hand.

Hope this helped.