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First Derivative Test Classification

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First Derivative Test Classification

Use the first derivative test to classify stationary points. Show enough working to make your reasoning clear. Unless stated otherwise, assume each function is differentiable on the interval considered.

First Derivative Test: Key Ideas

1.At a stationary point, the derivative changes from positive to negative as x increases. What type of stationary point is it?

A local maximum
A local minimum
A stationary point of inflection
The test cannot classify it

2.A function has derivative f′(x) = (x + 4)(x − 1)(x − 3). At which x-values can it have stationary points?

x = −4, 1 and 3
x = 4, −1 and −3
x = −4 only
x = 1 and 3 only

3.For a differentiable function, which change in the sign of f′(x) identifies a local minimum?

Positive to negative
Negative to positive
Positive on both sides
Negative on both sides

4.At x = 2, f′(2) = 0, and f′(x) is positive immediately to the left and right of 2. What does the first derivative test conclude?

There is a local maximum at x = 2
There is a local minimum at x = 2
The test does not classify x = 2 as a maximum or minimum
There cannot be a stationary point at x = 2

Find and Classify Stationary Points

5.For f(x) = x³ − 6x² + 9x + 1, find f′(x), then determine the coordinates of every stationary point.

6.Use the first derivative test to classify both stationary points of f(x) = x³ − 6x² + 9x + 1. Include the derivative signs on either side of each stationary x-value.

7.For g(x) = 2x³ − 12x + 4, find g′(x) and solve g′(x) = 0 to locate its stationary points.

8.For g(x) = 2x³ − 12x + 4, test the signs of g′(x) in the intervals x < −√2, −√2 < x < √2, and x > √2. Use these signs to classify the stationary points.

Apply the Test in Context

9.A small business models its weekly profit, in thousands of dollars, by P(t) = −2t³ + 15t² − 24t + 50, where t is the number of weeks after a new product launches. Find P′(t) and the times when the rate of change of profit is zero.

10.For P(t) = −2t³ + 15t² − 24t + 50, use the first derivative test to classify the stationary points and interpret them in terms of the business's profit.

11.The derivative of h is h′(x) = (x + 2)(x − 1)(x − 3). Draw a sign chart for h′(x), then classify the stationary points at x = −2, x = 1 and x = 3.