Everything You Need to Know About Sketching Curves Using Derivatives

The first and second derivative tests are pivotal tools in calculus for analyzing the behavior of functions. They assess where functions increase or decrease, identify local extrema, and determine concavity. While the first derivative reveals increasing or decreasing trends and locates potential extrema, the second derivative offers insights into the function's concavity, indicating whether a point is a local maximum or minimum. It also identifies inflection points where concavity changes. This comprehensive approach is essential, especially when derivatives present complexities, such as being zero or undefined, necessitating further exploration to fully understand the function's nature and behavior.

Overview:

The first derivative test involves examining where the derivative changes sign to identify local maxima and minima: a function is increasing where its derivative is positive and decreasing where it’s negative.

The second derivative test further refines this analysis. A positive second derivative implies the graph is concave up, suggesting a local minimum; a negative second derivative indicates concave down, pointing to a local maximum.

Additionally, points where the second derivative changes sign are inflection points, marking transitions between concave up and down. Together, these tests provide a comprehensive toolkit for analyzing a function’s behavior and identifying key features of its graph.

Except for situations where the first derivative is zero or undefined and the second derivative test is inconclusive. In such cases, further analysis or alternative methods, like the first derivative test or examining function values around the critical point, might be needed to accurately determine the nature of the extrema. Additionally, if the second derivative is zero or undefined at a point, it may not conclusively indicate an inflection point, requiring additional investigation.

(An inflection point is where a function’s graph changes concavity, shifting from concave up to concave down, or vice versa.)

Example:

$$\text{Analyze and sketch the curve of } f(x) = x^3 – 3x^2 – 9x + 27.$$

First Derivative (for critical points and slope):

$$f(x) = x^3 – 3x^2 – 9x + 27$$

$$f'(x) = 3x^2 – 6x – 9$$

Find Critical Points($$x$$ points where first derivative is zero):

$$3x^2 – 6x – 9 = 0$$

After solving this using delta method ($$[(b^2)-4ac]$$, we get $$3$$ and $$-1$$ as the results. At these points, first derivative is zero.

So it’s decreasing at $$x=-1$$ until it reaches $$3$$, [meaning it was increasing before $$-1$$

and it’s increasing from $$x=3$$.

Second Derivative (for concavity and points of inflection):

$$f”(x) = 6x – 6$$

$$6x – 6 = 0$$

So at $$x=1$$, concavity changes.

Let’s take a look at the functions graph:

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