FINDING CRITICAL NUMBERS OF A FUNCTION WORKSHEET

Find any critical numbers of the function.

Problem 1 :

f(x) = x³ - 3x²

Solution

Problem 2 :

g(x) = x⁴ - 4x²

Solution

Problem 3 :

g(t) = t√(4 - t), t < 3

Solution

Problem 4 :

Solution

Problem 5 :

h(x) = sin²x + cos x

0 < x < 2π

Solution

Problem 6 :

f(θ) = 2 sec θ + tan θ 

0 < θ < 2π

Solution

Use Calculus to find the CRITICAL POINTS of each of the following functions:

Problem 1 :

y = x² - 6x + 5

Solution

Problem 2 :

y = x³/(x² - 1)

Solution

Problem 3 :

f(x) = csc x ; [-π, π]

Solution

Problem 4 :

f(x) = e^x - x

Solution

Problem 5 :

f(x) = 6x⁵ + 33x⁴ - 30x³ + 100

Solution

Problem 6 :

f(x) = 4x³ - 9x² - 12x + 3

Solution

Problem 7 :

f(x) = 2/(t² - 4)

Solution

Problem 8 :

f(x) = (ln x)²

Solution

Problem 9 :

f(x) = 2 sin(x/2) where [-2π, 2π]

Solution

Problem 10 :

f(x) = x² ln (x)

Solution

Problem 11 :

The first derivative of the function 𝑓 is given by

f'(x) = sin²x/9 - (2/9)

How many critical values does f on the open interval (0, 10) ?

Solution

Problem 12 :

If 𝑓 is a continuous, decreasing function on [0,10] with a critical point at (4, 2), which of the following statements must be false?

(A) 𝑓(10) is an absolute minimum of f on [0,10].

(B) 𝑓(4) is neither a relative maximum nor a relative minimum.

(C) 𝑓′(4) does not exist

(D) 𝑓′(4) = 0            (E) 𝑓'(4) < 0

Solution