# First Derivative Test vs Second Derivative Test for Local Extrema

When determining local extrema in calculus, two key tests are utilized: the First Derivative Test and the Second Derivative Test. The First Derivative Test examines the sign changes of the derivative around a critical point to determine if it is a local minimum or maximum. In contrast, the Second Derivative Test involves analyzing the concavity of the function at the critical point; a concave-upward shape indicates a local minimum, while a concave-downward shape indicates a local maximum. These tests are fundamental in analyzing the behavior of functions and identifying critical points in mathematical contexts.

Questions

- How do you find the exact relative maximum and minimum of the polynomial function of #f(x) = x^3 + x + 4/x#?
- How do you find the exact relative maximum and minimum of the polynomial function of #p(x) =80+108x-x^3 #?
- How do you find the second derivative test to find extrema for #f(x) = 2x^2lnx-5x^2#?
- How do you find the exact relative maximum and minimum of the polynomial function of #f(x) = 2 csc x#?
- Find the maximum and minimum values for the function #f# defined by #f(x) = 2sinx + cos2x# in the interval #[0, pi/2]#?
- How do you find the exact relative maximum and minimum of the polynomial function of #f(x) =2x^3-3x^2-12x#?
- How do you know when to use the first derivative test and the second derivative test?
- How do you find the exact relative maximum and minimum of the polynomial function of #4x^8 - 8x^3+18#?
- How does the first derivative test work?
- How do you find the exact relative maximum and minimum of the polynomial function of #f(x)=(x^3)-12x+2#?
- What does the first derivative test tell you?
- How do you find the exact relative maximum and minimum of the polynomial function of #f(x) = x^3 - 2x^2 - x +1#?
- How do you find the exact relative maximum and minimum of the polynomial function of #f(x) = x^3-6x^2+15#?
- How do you find the exact relative maximum and minimum of the polynomial function of #f(x)=4x-x^2#?
- How do you find the exact relative maximum and minimum of the function of #f(x) = 4x+6x^-1#?
- How do you find the exact relative maximum and minimum of the polynomial function of #f(x)=x^3+6x^2-36x#?
- How do you use the second derivative test to find the local extrema for #f(x)=sin(x)#?
- How do you find the exact relative maximum and minimum of the polynomial function of #f(x)=x^3-3x+6#?
- What is the first derivative test for local extreme values?
- How do you find the exact relative maximum and minimum of the polynomial function of #2x^3 -23x^2+78x-72#?