first derivative test
states that if c is a critical point of f, and f is continuous and differentiable on an open interval containing c (except possible at c), then f(c) can be classified as follows:1) If f'(c)<0 for x0 for s>c, then f(c) is a relative minimum of f2) if f'(c)>0 for xc, then f(c) is a relative maximum of f
The first derivative test is a method used to determine the extrema (maxima and minima) points of a function. It uses the first derivative of the function and its sign changes to identify the critical points and the nature of the extrema.
To perform the first derivative test:
1. Find the critical points of the function by setting the derivative equal to zero and solving for x.
2. Determine the sign of the derivative on either side of each critical point.
a. If the sign of the derivative changes from negative to positive (i.e., the function is increasing), the critical point is a local minimum.
b. If the sign of the derivative changes from positive to negative (i.e., the function is decreasing), the critical point is a local maximum.
c. If the sign of the derivative does not change (i.e., the function is not increasing or decreasing) at the critical point, then it is not an extrema point.
3. Determine the nature of the extrema by examining the second derivative of the function at each critical point.
a. If the second derivative is positive at the critical point, then it is a local minimum.
b. If the second derivative is negative at the critical point, then it is a local maximum.
c. If the second derivative is zero at the critical point or undefined, then the first derivative test is inconclusive, and other methods should be used to determine the nature of the extrema.
Overall, the first derivative test is a quick and efficient method for determining the local extrema of a function. However, it is essential to remember that it only provides information about the local extrema and not about the global behavior of the function.
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The first derivative test is a method used to determine the extrema (maxima and minima) points of a function. It uses the first derivative of the function and its sign changes to identify the critical points and the nature of the extrema.
To perform the first derivative test:
1. Find the critical points of the function by setting the derivative equal to zero and solving for x.
2. Determine the sign of the derivative on either side of each critical point.
a. If the sign of the derivative changes from negative to positive (i.e., the function is increasing), the critical point is a local minimum.
b. If the sign of the derivative changes from positive to negative (i.e., the function is decreasing), the critical point is a local maximum.
c. If the sign of the derivative does not change (i.e., the function is not increasing or decreasing) at the critical point, then it is not an extrema point.
3. Determine the nature of the extrema by examining the second derivative of the function at each critical point.
a. If the second derivative is positive at the critical point, then it is a local minimum.
b. If the second derivative is negative at the critical point, then it is a local maximum.
c. If the second derivative is zero at the critical point or undefined, then the first derivative test is inconclusive, and other methods should be used to determine the nature of the extrema.
Overall, the first derivative test is a quick and efficient method for determining the local extrema of a function. However, it is essential to remember that it only provides information about the local extrema and not about the global behavior of the function.
More Answers:
The Power Of The Greatest Integer Function In Mathematics: Properties And ApplicationsMastering Cubic Functions: Graph Shapes And Solving For Roots
Unlocking The Mysteries Of The Square Root Function: Properties And Applications