Let’s talk about the first derivative test, a super handy tool in calculus for analyzing functions! It helps us find out where a function is increasing or decreasing and also helps identify its maximum and minimum values (called extremum points). So, how does it work? The first derivative test uses the function’s first derivative to figure out critical points, which are places where the function might change direction.

Here’s the gist: If a function is increasing on an interval, it means that as you move from left to right, the values of the function keep getting bigger. On the other hand, if the function is decreasing, it means the values drop as you move from left to right.

This is really important for exams like the SAT, ACT, or AP Exams, especially when you’re asked to find the intervals where a function is increasing or decreasing. Understanding the first derivative test can make these types of questions way easier to solve. So, if you ever see questions about functions in calculus, this is the go-to method for breaking them down!

In other words, increasing means the function is moving upward, and decreasing implies the function is moving downward.

Using First Derivative Test to Find Local Maxima and Minima

The quickest and easiest way to determine a function’s local maximum and minimum points is to use the first derivative test. The first derivative test relies on the idea of approximation to determine the local maxima and local minima by taking the point to the left and to the right in the vicinity of the critical points and substituting it in the first derivative formula.

Let c be a critical number for a continuous function f

If f′(x) changes from positive to negative at c, then f(c) is a local maximum.

If f′(x) changes from negative to positive at c, then f(c) is a local minimum.

If f′(x) does not change sign at c, then f(c) is neither a local maximum nor a minimum.

A function f has a local maximum or relative maximum at a point c if the values f(x) of f for x ‘near’ c are all less than f(c). Thus, the graph of f near c has a peak at c. A function f has a local minimum or relative minimum at a point c if the values f(x) of f for x ‘near’ c are all greater than f(c). Thus, the graph of f near c has a trough at c.

Thus we can conclude that, if these values are both less than the value of the function at the critical point, we can deduce that the critical point is a local maximum. If they are both bigger than the value at that point of the function, then the critical point must be a local minimum.

Local Maxima

A local maximum point on a function is a point (x,y) on the graph of the function whose y coordinate is larger than all other y coordinates on the graph at points “close to” (x,y). More precisely, (x,f(x)) is a local maximum if there is an interval (a,b) with a<x<b and f(x)≥f(z) for every z in both (a,b) and the domain of f.

Local Minima

(x,y) is a local minimum point if it has locally the smallest yy coordinate. Again being more precise: (x,f(x)) is a local minimum if there is an interval (a,b) with a<x<b and f(x)≤f(z) for every z in both (a,b) and the domain of ff. A local extremum is either a local minimum or a local maximum.

Necessary Conditions for Minima and Maxima

Only those values of the variables for which the n partial derivatives either disappear simultaneously (stationary points) or at which one or more of these derivatives cease to exist, determine the maximum or minimum of a continuous function of n variables in the interior of a region (i.e., are discontinuous).

Sufficient Conditions for Minima and Maxima

As we’ve shown, saddle or inflection points can also exist at stationary places, thus they’re not necessarily local maxima and minima. Procedures must now be developed to determine if stationary sites are maxima or minima. The construction of these essential conditions will begin with one independent variable, and they will then be enlarged using the same ideas for two and n independent variables. Once the local maxima and minima have been determined, the global maximum and minimum can be determined by comparing the various sites.

Point {x}={c} is global or absolute maximum (minimum) of function {y}={f{{\left({x}\right)}}} if {f{{\left({c}\right)}}}\ge{f{{\left({x}\right)}}}\left({f{{\left({c}\right)}}}\le{f{{\left({x}\right)}}}\right) for all {x} from the domain of {f{}}.

The greatest (smallest) value that the function can accept across its scope is known as the global maximum (minimum).

The point is or is not an extreme point depending on whether the first non-vanishing derivative is of even or odd order at a stationary point where the first and perhaps some of the higher derivatives disappear. If it’s even, the derivative maximum or minimum depends on whether it’s negative or positive.

Steps for Applying First Derivative Test

The subsequent series of steps can be used to perform the first derivative test.

• Find the first derivative of the given function f(x) and the function itself (x).
• Equalising the first derivative expression to zero will help you locate the limiting points.
• Find a point on the limiting point’s surrounding right and left sides, and then substitute the first derivative function there.
• The local maxima serves as the limiting point if the derivative of the function for the surrounding point to the left is positive f(x) > 0, and if the neighbouring point to the right is negative f(x) 0.
• The local minima serve as the limiting point if the derivative of the function for the neighbouring point on the left is negative f(x) 0, and for the neighbouring point on the right it is positive f(x) > 0.

Uses of First Derivative Test

The uses of first derivative test are as follows:

• The first derivative test is used to determine the local maxima and minima of a function as well as to determine where a function is increasing or decreasing on its domain. Thus, it reveals the precise location of the function’s minimum.
• A critical point of a function can be tested using the first derivative to see if it is a local maximum, local minimum, or neither of these.
• The slope of the line tangent to the graph of a function at a certain point is the first derivative.

Solved Example on First Derivative Test

Now that we have learnt everything about the first derivative test, let’s apply it to some functions. Here are some solved examples on the first derivative test.

Solved Example 1: Find where the function f(x) = x^3 – 12x – 5 is increasing and where it is decreasing.

Solution:

f(x) = x^3 – 12x – 5

Step 1: Find the derivative of f(x)

f ’(x) = 3x^2 – 12 = 3(x^2 – 4) = 3(x – 2)(x + 2)

Step 2: Equate f ’(x) = 0 to get the critical numbers. We achieve this by factoring the polynomials.

f ’(x) = 3(x – 2)(x + 2) = 0

x = 2, –2

Step 3: Determine the sign of f ‘(x) for each interval by setting up intervals with the essential integers as their endpoints. For each interval, use the increasing/decreasing test to determine if f(x) is increasing or decreasing.

Solved Example 2: Use the first derivative test to find the location of all local extrema for f(x)=x^3−3x^2−9x−1. Use a graphing utility to confirm your results.

Solution:

f(x)=x^3−3x^2−9x−1

Step 1: Find the derivative of f(x)

f′(x)=3x^2−6x−9.

Step 2: Equate f ’(x) = 0 to get the critical numbers. We achieve this by factoring the polynomials.

f′(x)=0.

3(x^2−2x−3)=3(x−3)(x+1)=0.

Therefore, the critical points are x =3,−1. Now divide the interval (−\infty, \infty) into the smaller intervals (−\infty, −1),(−1, 3) and (3,\infty).

Step 3: Determine the sign of f ‘(x) for each interval by setting up intervals with the essential integers as their endpoints. For each interval, use the increasing/decreasing test to determine if f(x) is increasing or decreasing. For example, let’s choose x = −2, x = 0, and x = 4 as test points.

Conclusion

In conclusion, the first derivative test is a powerful tool for analyzing functions in calculus. It helps identify where a function is increasing or decreasing, as well as locate its local maxima and minima. By understanding how the first derivative behaves around critical points, you can quickly determine the function’s behavior. This knowledge is essential for exams like the SAT, ACT, AP, and others, making calculus problems much easier to tackle!