Calculus: Minimums and Maximums of Functions

A point where the function changes from decreasing to increasing = Local minimum A point where the function changes from increasing to decreasing = Local maximum A point where the derivative is equal to 0 or undefined = Critical point A point where the function changes from decreasing to increasing or increasing to decreasing = Saddle point

If f'(a) changes sign from negative to positive at x=a, then f(x) has a local minimum at x=a = First Derivative Test If f''(a) > 0, then f(x) has a local minimum at x=a = Second Derivative Test If f'(a) does not change sign at x=a, then f(x) has a saddle point at x=a = First Derivative Test If f'(a) changes sign from positive to negative at x=a, then f(x) has a local maximum at x=a = First Derivative Test

The smallest value of f(x) over its entire domain = Global minimum The largest value of f(x) over its entire domain = Global maximum A point where the function changes from decreasing to increasing = Local minimum A point where the function changes from increasing to decreasing = Local maximum

<p>Finding the minimum or maximum of a function to solve real-world problems = Optimization Describing the behavior of physical systems = Physics and engineering Determining the critical points of a function = Calculus Graphing functions to visualize their behavior = Graph theory</p> Signup and view all the answers

<p>f''(a) &gt; 0 = Local minimum f''(a) &lt; 0 = Local maximum f'(a) = 0 = Critical point f'(a) changes sign from negative to positive at x=a = Local minimum</p> Signup and view all the answers

<p>A point where the derivative is equal to 0 or undefined = Critical point A point where the function changes from decreasing to increasing = Local minimum The smallest value of f(x) over its entire domain = Global minimum The largest value of f(x) over its entire domain = Global maximum</p> Signup and view all the answers

<p>A point where the function changes from increasing to decreasing = Local maximum A point where the function changes from decreasing to increasing = Local minimum The largest value of f(x) over its entire domain = Global maximum The smallest value of f(x) over its entire domain = Global minimum</p> Signup and view all the answers

<p>First Derivative Test = Determining the type of critical point Second Derivative Test = Determining the type of local extrema Global minimum and maximum = Optimization problems Critical points = Calculus</p> Signup and view all the answers

<p>A point where the derivative is equal to 0 or undefined = Critical point A point where the function changes from decreasing to increasing = Local minimum The smallest value of f(x) over its entire domain = Global minimum The largest value of f(x) over its entire domain = Global maximum</p> Signup and view all the answers

<p>If f'(a) changes sign from negative to positive at x=a, then f(x) has a local minimum at x=a = First Derivative Test If f''(a) &gt; 0, then f(x) has a local minimum at x=a = Second Derivative Test If f'(a) does not change sign at x=a, then f(x) has a saddle point at x=a = First Derivative Test If f'(a) changes sign from positive to negative at x=a, then f(x) has a local maximum at x=a = First Derivative Test</p> Signup and view all the answers