Explain how to apply the First Derivative Test.

Question

accimaroyalde · Accepted Answer

The first derivative test is the process of analyzing functions using their first derivatives in order to find their extremum point. This process is called the First derivative test.
Now if f'(x) changes positive to negative at the critical point then local maximum and if the derivative of the function changes negative to positive at the critical point the minimum and if f'(x) does not change signs then the function is neither maxima nor minima at that critical point.

Joseph Lewis · Accepted Answer

The second derivative test is used to find the relative maxima or relative minima.
The second derivative test:
Suppose the function f(x) has critical point at x = c that is f'(c) = 0 and f''(x) is continuous in the neighborhood of c, then
1. If f''(c) < 0 then f(x) has relative minima at x = c.
2. If f''(c) = 0 then f(c) can be relative minima or relative maxima or neither.
Steps to apply the second derivative test:
1. Find all critical points of function y= f(x) that is the points at which f'(x) = 0 or f'(x) does not exist.
2. Compute the second derivative of f(x) at those critical points.
3. a) If the second derivative at some critical point is positive then at that point the function f(x) has relative maxima.
b) If the second derivative at some critical point is negative then at that point the function f(x) has relative minima. and
c) If second derivative is zero at some critical point then test is inconclusive.

Explain how to apply the First Derivative Test.

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