(a) find the critical numbers of f (if any), (b) find the...

(a) find the critical numbers of $f$ (if any), (b) find the open interval(s) on which the function is increasing or decreasing, (c) apply the First Derivative Test to identify all relative extrema, and (d) use a graphing utility to confirm your results. $$f(x)=\left\{\begin{array}{ll} -x^{3}+1, & x \leq 0 \\ -x^{2}+2 x, & x>0 \end{array}\right.$$

(a) find the critical numbers of $f$ (if any), (b) find the open interval(s) on which the function is increasing or decreasing, (c) apply the First Derivative Test to identify all relative extrema, and (d) use a graphing utility to confirm your results.
$$f(x)=\left\{\begin{array}{ll}
-x^{3}+1, & x \leq 0 \\
-x^{2}+2 x, & x>0
\end{array}\right.$$

Key Concepts

Increasing and Decreasing Intervals

Determining intervals of increase or decrease involves analyzing the sign of the derivative. Where the derivative is positive, the function is increasing; where it is negative, the function is decreasing. This analysis is a core tool in understanding the overall behavior of a function.

Critical Numbers

Critical numbers are values within the domain of a function where the derivative is zero or does not exist. These points are essential for analyzing the behavior of the function because they may indicate potential local extrema or points where the function's behavior changes.

First Derivative Test

The First Derivative Test is used to classify critical numbers as relative maxima, minima, or neither by examining the sign changes of the derivative around these points. It is an effective method to identify local extrema based on the behavior of the function's slope.

Graphing Utilities in Calculus

Graphing utilities serve as a powerful tool for visualizing functions and confirming analytical results. They are used to check the accuracy of derivative tests, identify trends, and provide intuitive insight into function behavior.

Piecewise Function Analysis

Analyzing piecewise functions requires careful consideration of the function's definition on different intervals, including points where the definition changes. This involves checking continuity and differentiability at the boundaries, which is essential to accurately applying derivative-based tests and understanding overall behavior.

Transcript

0:00 So this is s .m.

00:01 44 and we need to do find critical points.

00:07 If they define the interval and determine it is increasing or decreasing for that interval.

00:12 Lastly, determine extremer and use the graphic utility to confirm our answer.

00:17 So yeah.

00:18 So this is my question here and i will differentiate on both sides.

00:22 So as i differentiate, i have derivative of f and i will here have 3x squared for x less than equals or equals 0 and negative 2x for x greater than 0 negative 2x plus 2 sorry for x greater than 0 so now if i look closely my function has the derivative for x equals to 0 is not defined because when coming from left hand side the derivative is 0 but from where is derivative approaching from the right hand side is positive 2 which means the function is not differentiable for x equals to 0 so here goes my first point that is my derivative which is x equals to 0 now i need to equate my function which is minus 3x square equals to 0 but there's a mistake here here i will not include equal to sign because as i already told you that my derivative is not defined for x equals to 0 so we will remove our equals to sign here and we need to find the point for both functions where my derivative is 0 so here if i equate minus 3 x square for equal to 0 i will get a 0 but we already know that our function gradient is not defined for 0 so now we will equate this negative to x plus 2 equals to 0 and we get x equals to 1 so our second critical point is 1 so now after we have two critical points we need to define our intervals.

02:18 My first interval will be from negative infinity to 0, then from 0 to 1 to positive infinity.

02:28 Simply.

02:29 I'll have my test failure so that i can find out the signs of derivative of f...

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