How to Find the Minimum of a Function

Oct 13, 2025 by magicalceylontours.com 68 views

Introduction

In mathematics, particularly in calculus, understanding the behavior of functions is crucial. One of the key aspects of function behavior is identifying the location of their minimum or maximum values. In this article, we will explore how to determine the function with a minimum located at a specific point, in this case, (4, -3).

What is a Minimum?

A minimum of a function is the lowest point on its graph. It is the value of the function at a specific point where the function starts to increase. In other words, it is the point where the function has the lowest value.

How to Find the Minimum of a Function

To find the minimum of a function, we need to find the critical points of the function. Critical points are the points where the function changes from increasing to decreasing or vice versa. We can find the critical points by taking the derivative of the function and setting it equal to zero.

Derivatives and Critical Points

A derivative of a function is a measure of how the function changes as its input changes. It is denoted by the symbol f'(x). To find the critical points, we need to set the derivative equal to zero and solve for x.

Finding the Derivative

Let's find the derivative of each function given in the options.

Option A: $f(x) = -\frac{1}{2} x^2 + 4x - 11$

To find the derivative of this function, we will use the power rule of differentiation, which states that if f(x) = x^n, then f'(x) = nx^(n-1).

import sympy as sp

# Define the variable
x = sp.symbols('x')

# Define the function
f = -1/2*x**2 + 4*x - 11

# Find the derivative
f_prime = sp.diff(f, x)

print(f_prime)

The derivative of this function is f'(x) = -x + 4.

Option B: $f(x) = -2x^2 + 16x - 35$

To find the derivative of this function, we will use the power rule of differentiation.

import sympy as sp

# Define the variable
x = sp.symbols('x')

# Define the function
f = -2*x**2 + 16*x - 35

# Find the derivative
f_prime = sp.diff(f, x)

print(f_prime)

The derivative of this function is f'(x) = -4x + 16.

Option C: $f(x) = \frac{1}{2} x^2 - 4x + 5$

To find the derivative of this function, we will use the power rule of differentiation.

import sympy as sp

# Define the variable
x = sp.symbols('x')

# Define the function
f = 1/2*x**2 - 4*x + 5

# Find the derivative
f_prime = sp.diff(f, x)

print(f_prime)

The derivative of this function is f'(x) = x - 4.

Option D: $f(x) = 2x^2 - 16x + 35$

To find the derivative of this function, we will use the power rule of differentiation.

import sympy as sp

# Define the variable
x = sp.symbols('x')

# Define the function
f = 2*x**2 - 16*x + 35

# Find the derivative
f_prime = sp.diff(f, x)

print(f_prime)

The derivative of this function is f'(x) = 4x - 16.

Finding the Critical Points

Now that we have the derivatives of each function, we can find the critical points by setting the derivative equal to zero and solving for x.

Option A: $f(x) = -\frac{1}{2} x^2 + 4x - 11$

To find the critical points of this function, we will set the derivative equal to zero and solve for x.

import sympy as sp

# Define the variable
x = sp.symbols('x')

# Define the derivative
f_prime = -x + 4

# Set the derivative equal to zero and solve for x
critical_points = sp.solve(f_prime, x)

print(critical_points)

The critical point of this function is x = 4.

Option B: $f(x) = -2x^2 + 16x - 35$

To find the critical points of this function, we will set the derivative equal to zero and solve for x.

import sympy as sp

# Define the variable
x = sp.symbols('x')

# Define the derivative
f_prime = -4*x + 16

# Set the derivative equal to zero and solve for x
critical_points = sp.solve(f_prime, x)

print(critical_points)

The critical point of this function is x = 4.

Option C: $f(x) = \frac{1}{2} x^2 - 4x + 5$

To find the critical points of this function, we will set the derivative equal to zero and solve for x.

import sympy as sp

# Define the variable
x = sp.symbols('x')

# Define the derivative
f_prime = x - 4

# Set the derivative equal to zero and solve for x
critical_points = sp.solve(f_prime, x)

print(critical_points)

The critical point of this function is x = 4.

Option D: $f(x) = 2x^2 - 16x + 35$

To find the critical points of this function, we will set the derivative equal to zero and solve for x.

import sympy as sp

# Define the variable
x = sp.symbols('x')

# Define the derivative
f_prime = 4*x - 16

# Set the derivative equal to zero and solve for x
critical_points = sp.solve(f_prime, x)

print(critical_points)

The critical point of this function is x = 4.

Conclusion

$In conclusion, the function with a minimum located at (4, -3) is **Option A: $f(x) = -\frac{1}{2} x^2 + 4x - 11$**. This is because the critical point of this function is `x = 4`, which is the same as the x-coordinate of the given point.<br/> **Q&A: The Graph of Which Function Has a Minimum Located at (4, -3)?**$

Q: What is the minimum of a function?

A: The minimum of a function is the lowest point on its graph. It is the value of the function at a specific point where the function starts to increase.

Q: How do I find the minimum of a function?

A: To find the minimum of a function, you need to find the critical points of the function. Critical points are the points where the function changes from increasing to decreasing or vice versa. You can find the critical points by taking the derivative of the function and setting it equal to zero.

Q: What is the derivative of a function?

A: The derivative of a function is a measure of how the function changes as its input changes. It is denoted by the symbol f'(x).

Q: How do I find the derivative of a function?

A: To find the derivative of a function, you can use the power rule of differentiation, which states that if f(x) = x^n, then f'(x) = nx^(n-1).

Q: What is the power rule of differentiation?

A: The power rule of differentiation is a rule that states that if f(x) = x^n, then f'(x) = nx^(n-1).

Q: How do I find the critical points of a function?

A: To find the critical points of a function, you need to set the derivative of the function equal to zero and solve for x.

Q: What is the significance of the critical points of a function?

A: The critical points of a function are the points where the function changes from increasing to decreasing or vice versa. They are also the points where the function has a minimum or maximum value.

Q: How do I determine which function has a minimum located at (4, -3)?

A: To determine which function has a minimum located at (4, -3), you need to find the critical points of each function and compare them to the given point.

Q: What is the final answer?

A: The final answer is Option A: $f(x) = -\frac{1}{2} x^2 + 4x - 11$ .

Q: Why is Option A the correct answer?

A: Option A is the correct answer because the critical point of this function is x = 4, which is the same as the x-coordinate of the given point.

Q: What is the minimum value of the function?

A: The minimum value of the function is -3, which is the y-coordinate of the given point.

Q: How do I graph the function?

A: To graph the function, you can use a graphing calculator or a computer program such as Desmos or GeoGebra.

Q: What is the significance of graphing the function?

A: Graphing the function allows you to visualize the behavior of the function and see the location of the minimum value.

Q: How do I use the graph to determine the minimum value?

A: To use the graph to determine the minimum value, you can look for the lowest point on the graph, which will be the minimum value of the function.

Q: What is the final answer in terms of the graph?

A: The final answer is that the graph of Option A: $f(x) = -\frac{1}{2} x^2 + 4x - 11$ has a minimum located at (4, -3).