First and Second Derivative Tests
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First Derivative Test
If there is a critical number "c" for a continuous function, then 1) if f' changes from (+) to (-) at c, f has a local maximum at c. 2) If f' changes from (-) to (+) at c, then f has a local minimum at c. 3) If f' does not change sign at c, then f has no local maximum or minimum.
Second Derivative Test
If f is continuous by c, the 1) if the f'(c) is greater than zero, f has a local minimum at c 2) if f'(c)=zero and f(c) is less than zero, f has a local maximum at c.
Inflection point: A point on the graph where the curve changes from concave up to concave down.
Concavity Test: If f is greater than zero, f is concave up, if f is less than zero, f is concave down
EXAMPLE: Sketch the function y=x^{4} - 4x^{3}.
Step 1. Write the f' and f
Step 2. Find the critical points by setting f'=0
Step 3. Evaluate f" at critical points
Step 4. Determine the concavity:
If f is <0, it is concave down
If f is >0, it is concave up
A point where the concavity changes from up to down or vice-versa is called an inflection point
EXAMPLE 2: For the function, f(x)=4x^{3}-2x^{2}+1, find where the function is increasing/decreasing and the concavity.
Solution:
1. f'= 12x^{2}-4x , f"= 24x-4
2. Set f'=0 --> 12x^{2}-4x =0
3x^{2}-4x =0
x(3x-4)=0
x=0, x=4/3
3. Evaluate the f" at the critical points
f"(0)=24(0)-4= -4
f"(4/3)=24(4/3)-4= 28
4. Concavity