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NPTEL Data Science for Engineers Assignment 4 Answers 2023? In this article we will discuss about the answers for Week 4 assignment of Data science for Engineers. All these answers are make it as reference.I am confident in providing these answers.

Also Read: NPTEL Data Science for Engineers Assignment 3 Answers 2023  

About Nptel

National programme on Technology enhanced learning ( NPTEL ) , is a very best online platform to do internships. In this Platform, it is providing free internship to students and professionals in India. It is initiated taken by Ministry of Human Resource Development (MHRD) in the year 2003. It is collabarated with 7 IIT's and IIT sc . It is providing high quality education and cost effective education to students and the professionals across the country. It is providing free resource of learning, assignments and quizes to solve.   

NPTEL Data Science for Engineers Assignment 4 Answers 2023 

Last Date: 22-02-2023

You can find the answers for Data Science for Engineers Assignment 4 Answers 2023 below

Q1. Let f(x)=x^3+6x^2−3x−5. Select the correct options from the given options below : 

a. −2+√5 will give the maximum for f(x).

b. −2+√5 will give the minimum for f(x). 

c. The stationary points of f(x) are −2+√5 and −2+√5 

d. The stationary points of f(x) are -4 and 0

Answer: [ c , b ]  

               Reason : f '' (x) at  −2+√5 is > 0. Hence, it is called as local minima. 

Solution: 

             To find the stationary points of f(x), we need to find the values of x where f'(x) = 0.

f(x) = x^3 + 6x^2 - 3x - 5

f'(x) = 3x^2 + 12x - 3

Consider f'(x) = 0, we get:    3x^2 + 12x - 3 = 0

Dividing on both sides by 3:  x^2 + 4x - 1 = 0

Using the quadratic formula, we get:

x= -b ± sqrt(4*a*c - 2*a)

x = (-4 ± sqrt(16 + 4))/2 = (-4 ± sqrt(20))/2

x = -2 ± sqrt(5)

So the stationary points of f(x) are -2 + sqrt(5) and -2 - sqrt(5). 

To determine whether each point is a maximum or minimum, we need to look at the sign of f''(x) at each point.

f''(x) = 6x + 12

At x = -2 + sqrt(5), we have,

f''(-2 + sqrt(5)) = 6(-2 + sqrt(5)) + 12 = -6 + 6sqrt(5) > 0

So -2 + sqrt(5) is a local minimum.

At x = -2 - sqrt(5), we have,

f''(-2 - sqrt(5)) = 6(-2 - sqrt(5)) + 12 = -6 - 6sqrt(5) < 0

So -2 - sqrt(5) is a local maximum.

From the following information given below, answer the below questions Q2 and Q3.

Consider the following Optimization problem. 

                                                         maxxϵRf(x) , where
                                    f(x)=x^4+7x^3+5x^2−17^x+3

Let x∗ be the maximizer of f(x)

Q2. What is the second order sufficient condition for x∗ to be the maximize of the function f(x)? 

a. 4x^3 + 21x^2 + 10x - 17 = 0

b. 12x^2 + 42x + 10 = 0

c. 12x^2 + 42x + 10 > 0

d. 12x^2 + 42x + 10 < 0 

Answer: [ c]  12x^2 + 42x + 10 > 0 

Solution: 

                 

Q3. Find the value of x*

a. -4.48

b. 0.66

c. -1.43

d. 4.45

Answer: [ c ] -1.43  

Q4. Let f(x) = 2Sin x , 0≤x≤2π . Select the correct the options from the given following . 

a. π/2 is the global maximum of f(x).

b. π is the global minimum of f(x). 

c. 3π/2 is the global maximum of f(x). 

d. 3π/2 is the global maximum of f(x). 

Answer: [ a, d ]  

Using the following information answer the following questions Q5, Q6, Q7,Q8. 

Let f(x) = 2x1^2 + 3x1x2 + 3x2^2 + x1 + 3x2

Q5. Find the gradient of f(x). 

Answer: [ a ]  

Q6. Find the stationary point of f(x) 

a. 0.6, 0.4

b. -0.6, -0.4 

c. 0.2, -0.6

d. 0.2, 0.6

Answer: [ c ]   

   

Q7. Find the Hessian Matrix for f(x). 

                            

 

Answer: [ c ]  

Q8. The Stationary point Obtained in Q6 is a 

a. Maxima 

b. Minima 

c. Saddle Point 

Answer: [ b ]  

Q9. Let f(x1,x2)=4x1^2−4x1x2+2x2^2 . Select the correct options :

a. (2,4) is a stationary point of f(x). 

b. (0,0) is a stationary point of f(x). 

c. The Hessian matrix ▽2f is positive definite. 

d. The Hessian matrix ▽2f  is not positive definite. 

Answer: [ b, c ]  

Q10. In optimization problem, find the function that we want to optimize is called

a. Decision Function 

b. Constraints Function 

c. Optimal Function 

d. Objective Function 

Answer: [ d ]  

Solution: 

                   

Q11. The optimization problem min xf(x) can also be written as max xf(x). 

a. True 

b. False  

Answer: [ b ]  

Solution : 

              The optimization problem of minimizing a function f(x) can be written as:

          minimize f(x) = -maximize -f(x)

The problem of maximizing xf(x) is not equivalent to minimizing f(x) because the optimal value of x may differ between the two problems.

Q12. In the gradient descent algorithm, the step size should always be same for each iteration.

a. True 

b. False 

Answer: [ b ]  

Solution : 

               The step size in the gradient descent algorithm, also known as the learning rate, does not always have to be the same for each iteration. The choice of the learning rate can have a significant impact on the performance of the algorithm, and finding an appropriate learning rate can be challenging.

 

Also Read: NPTEL Data Science for Engineers Assignment 3 Answers 2023  

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