Name: finding turning points with the first derivative
Uploaded: 2026-01-12T12:24:39.000Z
Duration: 1 min 25 s
Channel: JoVE

Imagine an asset price that crashes to its lowest point, rebounds sharply as bargain-hunters step in, and then gradually declines.The plot&#8217;s high and low points, used in financial analysis, are identified using the first derivative test.To understand this, model the curve as a function, and then the first derivative is found by applying the product rule.The common terms are factored out, and each term is set to zero to get the critical points of the function, along with the corresponding intervals.After that, test points are selected in each interval, and then the sign of the function&#8217;s derivative is examined.A positive derivative shows that the function is increasing, while a negative derivative means that it is decreasing. When the derivative changes from positive to negative, the function shifts from increasing to decreasing, giving a local maximum. A change from negative to positive shows a local minimum.Substituting these x-values into the original function gives their corresponding function values, which are the local extrema.This gives the local extrema maxima and minima of the function, which are critical for analyzing asset valuation.

first derivative test: problem solving

First Derivative Test: Problem Solving

Imagine an asset price that crashes to a low point, rebounds sharply as bargain-hunters step in, and then gradually declines. Such behavior can be modeled with a smooth function whose turning points represent locally overvalued and undervalued regions. A convenient example that captures rebound followed by decay is:<figure class="image image-style-align-center" style="display:table;margin-left:auto;margin-right:auto;"><img style="aspect-ratio:300/24;" src="https://cloudfront.jove.com/files/ftp_upload/19829/19829_Equation_1_editor_19829.svg" alt="Mathematical function equation, \( f(x) = (x - 2)^4 e^{-x} \), exponential decay." data-altupdatedat="1775223722000" data-alt="19829_Equation_1.svg" /></figure>The high and low points of this curve are identified using the first derivative test, which determines where the function changes from increasing to decreasing or vice versa. To begin, the first derivative is computed. Because the function is a product of a polynomial term and an exponential term, differentiation requires the product rule. After differentiating, the resulting expression is simplified by factoring out terms common to all parts of the derivative.The derivative is then set equal to zero, and solving yields the critical points&#8212;inputs where the slope is zero or where the slope test must be checked. These critical points partition the domain into intervals for further analysis.Next, test points are chosen within each interval, and the sign of the derivative is evaluated. A positive derivative shows that the modeled asset price is increasing over that interval, while a negative derivative shows it is decreasing. A change in the derivative from positive to negative signals a transition from rising to falling, identifying a local maximum. A change from negative to positive identifies a local minimum, corresponding to a trough in the price curve.Finally, substituting the critical inputs back into the original function gives the corresponding price levels at these turning points. These local extrema are central to valuation analysis because they mark potential reversal regions and help quantify where momentum shifts from recovery to decline or from decline to recovery.

Imagine an asset price that crashes to a low point, rebounds sharply as bargain-hunters step in, and then gradually declines. Such behavior can be modeled ...

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JoVE Core

Calculus

Unit

Applications of Differentiation

KEY TERMS AND CONCEPTS

finding turning points with the first derivative

The first derivative test is used to find turning points on a smooth curve. It shows where a function changes from increasing to decreasing, or from decreasing to increasing. In this example, the curve can model an asset price that falls to a low point, rebounds, and then gradually declines again.

The process starts by finding the first derivative of the function. Because the function is a product of a polynomial term and an exponential term, the product rule is needed. After differentiating, the expression is simplified by factoring out the terms that are common to every part of the derivative.

Next, the derivative is set equal to zero. Solving this equation gives the critical points, which are inputs where the slope is zero or where the slope test must still be checked. These critical points split the domain into separate intervals for further analysis.

Test points are then chosen from each interval, and the sign of the derivative is checked. A positive derivative means the function is increasing on that interval. A negative derivative means the function is decreasing. When the sign changes from positive to negative, the curve has a local maximum. When the sign changes from negative to positive, the curve has a local minimum.

Finally, the critical inputs are substituted back into the original function to find the corresponding output values at those turning points. These local extrema are important in valuation analysis because they show where momentum changes from recovery to decline, or from decline to recovery. They help identify possible reversal regions on the curve.

Finding Turning Points with the First Derivative

finding-turning-points-with-the-first-derivative

absolute and local extreme values

Finding Max and Min on Function Graphs

Absolute and local extreme values show the highest and lowest points on a function graph. Absolute maximum and absolute minimum values describe the highest and lowest values across the entire domain. Local maximum and local minimum values describe the highest and lowest values within a smaller part of the graph. Periodic functions such as sine and cosine can have extreme values at many points. Their repeating pattern creates multiple local extrema. In these cases, the derivative becomes zero...

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critical-numbers-and-the-closed-interval-method

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first derivatives and the shape of a graph

first-derivatives-and-the-shape-of-a-graph

Using First Derivatives to Find Graph Changes

The first derivative shows how a function changes across its domain. In calculus, the derivative is written as f'(x). It tells us whether the graph is rising, falling, or turning at a given point. A function is increasing on an interval when its derivative is positive. That means the slope of the tangent line is positive. A function is decreasing on an interval when its derivative is negative. In that case, the tangent line slopes downward. These sign changes help describe the overall shape...

The first derivative test is used to find turning points on a smooth curve. It shows where a function changes from increasing to decreasing, or from decreasing to increasing. In this example, the curve can model an asset price that falls to a low point, rebounds, and then gradually declines again. The process starts by finding the first derivative of the function. Because the function is a product of a polynomial term and an exponential term, the product rule is needed. After differentiating,...

second derivatives and the shape of a graph

second-derivatives-and-the-shape-of-a-graph

Using Second Derivatives to Read Curvature

The second derivative shows how a graph bends. It gives key information about curvature and how that curvature changes over an interval. These ideas help with graph analysis in math and in real-world settings such as road elevation, population growth, and economic trends. A function f(x) is concave upward on an interval when its graph lies above all of its tangent lines. In this case, the second derivative is positive. That means the slope of the tangent line is increasing, and the graph bends...

second derivative test: problem solving

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slant asymptotes

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optimization problems

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application-of-differentiation-to-business

Maximizing Bakery Revenue with Calculus

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Antiderivative graphs show how a function accumulates over time. In calculus, an antiderivative tracks that buildup, much like the changing position of a rolling ball as its velocity changes. The graph of the antiderivative reflects the total effect of the original function's values. The sign of the original function helps determine the antiderivative's direction. When the function is positive, the antiderivative rises because the accumulation grows. When the function is negative, the...

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Finding Car Stopping Distance with Integration

Integration connects acceleration, velocity, and displacement in linear motion. It works as the reverse of a derivative, which tells how one quantity changes with another. In physics, this makes it possible to move from a rate of change back to the motion itself. A car provides a clear example. It is moving at a steady 20 meters per second when an obstacle appears 800 meters ahead. To stop in time, the car slows with a constant acceleration. That acceleration tells how the car’s velocity...

key terms and concepts

Use the first derivative test to find critical points, test intervals, and identify local maxima and minima on a smooth curve with an asset price example.

Finding Turning Points with the First Derivative | JoVE K12

The JoVE Core Calculus collection provides a comprehensive visual guide to the entire calculus sequence, seamlessly connecting mathematical theory to real-world applications in physics and economics. Through dynamic animated lessons, the course transitions smoothly from foundational single-variable concepts to advanced multivariable topics. Students will explore everything from standard differentiation and integration to spatial vectors, multi-dimensional derivatives, and complex vector fields. By breaking down abstract coordinate systems, infinite series, and spatial motion, this structured approach builds the robust analytical skills and advanced mastery essential for STEM success.

calculus

Finding Turning Points with the First Derivative

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