Lesson: First Derivative Test for Finding Local Extrema

In this lesson, we will learn how to find the local extrema using the first derivative test.

Worksheet: First Derivative Test for Finding Local Extrema • 13 Questions

Q1:

Find the point ( π‘₯ , 𝑓 ( π‘₯ ) ) where 𝑓 ( π‘₯ ) = | 9 π‘₯ + 9 | βˆ’ 5 has a critical point, and determine whether it is a local maximum or local minimum.

Q2:

Determine where the local maxima and minima are for 𝑓 ( π‘₯ ) = π‘₯ 4 βˆ’ 2 π‘₯ + 5 4 2 .

Q3:

Find the local maximum and minimum values of 𝑓 ( π‘₯ ) = βˆ’ 5 + 2 π‘₯ βˆ’ 2 π‘₯ 2 .

Q4:

Locate and classify the critical points of 𝑓 ( π‘₯ ) = 5 π‘₯ π‘₯ + 1 6 2 .

Q5:

Find, if any, the local maximum and local minimum values of 𝑓 ( π‘₯ ) = βˆ’ 2 4 π‘₯ + 1 7 π‘₯ l n .

Q6:

Let 𝑓 ( π‘₯ ) = 4 π‘₯ βˆ’ 2 4 π‘₯ c o s s i n 2 , find where the local maxima and local minima occur for 0 < π‘₯ < πœ‹ 2 .

Q7:

Identify and classify the critical points of 𝑓 ( π‘₯ ) = 2 𝑒 βˆ’ 5 π‘₯ 2 as local maxima, local minima, or neither.

Q8:

Find (if any) the local maxima and local minima of 𝑓 ( π‘₯ ) = 2 π‘₯ √ βˆ’ π‘₯ + 2 .

Q9:

Find the local maximum and minimum values of 𝑓 ( π‘₯ ) = 2 π‘₯ 3 π‘₯ 2 l n .

Q10:

Find, if any, the local maximum and local minimum values of 𝑓 ( π‘₯ ) = βˆ’ 5 𝑒 ( π‘₯ βˆ’ 7 ) π‘₯ .

Q11:

Find the local maximum and local minimum values of 𝑓 ( π‘₯ ) = π‘₯ ( βˆ’ 6 π‘₯ + 7 ) 2 3 . Give your answers to the nearest integer.

Q12:

Find the point ( π‘₯ , 𝑓 ( π‘₯ ) ) where 𝑓 ( π‘₯ ) = βˆ’ | βˆ’ 4 π‘₯ | + 4 has a critical point, and determine whether it is a local maximum or local minimum.

Q13:

Find the critical point π‘Ž of 𝑓 ( π‘₯ ) = 4 π‘₯ + 2 4 π‘₯ c o s c o s 2 that satisfies 0 < π‘Ž < πœ‹ 2 , and then state whether ( π‘Ž , 𝑓 ( π‘Ž ) ) is a local maximum or a local minimum.

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