To find the critical value of a function we simply take its derivative, set it to zero, and solve for x. The values of x for which the derivative is zero are critical numbers.

In this article, we will cover critical numbers and how to find critical numbers of function. We will also solve some problems related to how to find critical numbers of a function.

What are Critical Numbers of a Function?

The value of the variable present in the function obtained by equating its first derivative to zero is called the critical number of a function. In other words, the numbers for which the derivative of the function becomes zero are referred to as the critical numbers of a function.

The critical number of a function is used in analyzing the function’s behavior, such as determining where it has relative maxima, minima, or points of inflection.

How to Find Critical Numbers of a Function

To find the critical numbers of a function we follow the steps given below:

• First find the first derivative of the function.

• Then, equate the first derivative of a function {f′(x)} to zero.

• Then, solve the obtained equation and find the values of given variable.

• The result gives the critical numbers of a function.

Solved Examples on Critical Numbers of a Function

Example 1: Find the critical numbers of the function f(x) = x² + 2x.

Solution:

First we will find the first derivative of f(x) i.e., f'(x)

f'(x) = 2x + 2

Now equate f'(x) to zero

2x + 2 = 0

Now solve the above equation and obtain value of x.

2x = -2

x = -1

So, the critical number for the given function f(x) is -1.

Example 2: Find the critical numbers of the function f(x) = (x³ / 3) – 2x² + 4x + 1.

Solution:

First we will find the first derivative of f(x) i.e., f'(x)

f'(x) = 3(x² / 3 ) – 4x + 4

f'(x) = x² – 4x + 4

Now equate f'(x) = 0

x² – 4x + 4 = 0

(x – 2)² = 0

x = 2

So, the critical number of the function f(x) is 2.

Example 3: Find the critical numbers of function p(x) = x ln x.

Solution:

First we will find the first derivative of p(x) i.e., p'(x)

To differentiate p(x), we will use the product rule, which states that if u(x) and v(x) are functions of x, then (uv)′=u′v + uv′

Here, u(x) = x and v(x) = ln⁡x

u′(x) = 1 and v'(x) = 1/x

Using the product rule we have:

p′(x) = (x)′ln⁡x + x(ln⁡x)

p′(x) = 1 ⋅ ln⁡x + x ⋅ 1/x

p'(x) = ln x + 1

Now equate p'(x) = 0

1 + ln x = 0

ln x = -1

x = e^-1

The critical number of p(x) is e^-1 .

Example 4: Determine the critical number of the function a(x) = sin 2x.

Solution:

First, we will find a'(x)

a'(x) = 2 cos 2x

Now put a'(x) = 0

2 cos 2x = 0

cos 2x = 0

2x = π / 2

x = π / 4

Example 5: Find the critical number of f(x) = 2 sin x – x

Solution:

First, find f'(x)

f'(x) = 2cos x – 1

Now put f'(x) = 0

2 cos x – 1 = 0

2 cos x = 1

cos x = 1 / 2

x = π / 3

Example 6: Find the critical number of function f(x) = x³ – 3x² + 2x

Solution:

First find f'(x)

f'(x) = 3x² – 6x + 2

Now put f'(x) = 0

3x² – 6x + 2 = 0

Solve this quadratic equation:

[Tex]x = \frac{6\pm\sqrt{(-6)^2 – 4\cdot3\cdot2)}}{2\cdot3}[/Tex]

[Tex]x = \frac{6 \pm\sqrt{36-24}}{6}[/Tex]

[Tex]x = \frac{6 \pm\sqrt{12}}{6}[/Tex]

[Tex]x = \frac{6 \pm2\sqrt3}{6}= 1 \pm \frac{\sqrt{3}}{3}[/Tex]

So, the critical numbers are:

[Tex]x = 1 + \frac{\sqrt 3}{3}\ and \ 1 – \frac{\sqrt 3}{3} [/Tex]

Example 7: Find the critical numbers of function f(x) = (x/ 2) + (2 / x)

Solution:

f(x) = (x / 2) + (2 / x)

First find f'(x)

f'(x) = 1/2 + 2 (-1/x²)

f'(x) = 1/2 – 2/x²

Now put f'(x) = 0

1/2 – 2/x² = 0

1/2 = 2/x²

x² = 4

x = 2, -2

Example 8: Find the critical numbers of function f(x) = tan x – 2x

Solution:

First find f'(x)

f'(x) = sec² x – 2

Now, put f'(x) = 0

sec² x – 2 = 0

sec² x = 2

Taking root both sides

sec x = √2

x = π / 4 or 3π / 4

Example 9: Find the critical numbers of function f(x) = sin x + cos x

Solution:

f(x) = sin x + cos x

First find f'(x)

f'(x) = cos x – sin x

Now put f'(x) = 0

cos x – sin x = 0

Divide both sides by cos⁡x:

1 = tan x

x = π / 4 + kπ where k is any integer.

Example 10: Find the critical numbers of function f(x) = sin x + (1/2) cos x

Solution:

First find f'(x)

f'(x) = cos x – (1/2)sin x

Now, put f'(x) = 0

cos x – (1/2)sin 2x = 0

cos x = (1/2) sin x

Divide both sides by cos⁡x:

1 = (1/2)tan x

tanx = 2

The general solution for tan⁡x = 2 is:

x = tan⁡⁻¹(2) + kπ, where k is any integer.

Read More,

• How to Find Maximum and Minimum Values of a Function
• How to find the Equation of a Quadratic Function
• How to Find Rank of a 3×3 Matrix

Practice Questions on Critical Numbers of a Function

Q1. Find the critical values of function p(x) = 9x – 45.

Q2. Find the critical values of function f(x) = x³ + 3x² – 18x + 2.

Q3. Find the critical numbers of function f(x) = x³ lnx.

Q4. Find the critical numbers of function f(x) = cos x.

Q5. Find the critical numbers of function f(x) = sin 2x – x

Q6. Find the critical numbers of function f(x) = x √(2x – 1)

Q7. Find the critical numbers of function f(x) = x³ (2x – 1)³

Q8. Find the critical numbers of function f(x) = sin⁴x + cos⁴x

Q9. Find the critical numbers of function f(x) = x + a²/x

Q10. Find the critical numbers of function f(x) = log x / x

FAQs on Critical Numbers of a Function

How do You Find the Critical Values of a Function?

To find the critical values of a function f(x), we first find the first derivative of function, equate it to zero and solve for x.

What is the Rule for Critical Numbers?

The rule for critical number is that it makes the derivative of the function zero.

Does Every Function have a Critical Number?

No, every function does not have a critical number.

What is the Formula for Critical Value?

The formula for critical value of function f(x) is:

f'(x) = 0 (solve for x)

What are the Steps to Find Critical Number of the Function?

The steps to find critical number of the function are as follows:

• Find first derivative of function
• Equate first derivative of function to zero
• Now, solve for the given variable