The function 𝑓(𝑥)=−3𝑥4+16𝑥3+24𝑥2−192𝑥+10 has critical points at some of the following 𝑥 -values. In each case identify whether the point is a critical point, and if so find out if it is a maximum, minimum or neither maximum nor minimum. The point 𝑥=−6 is Answer 1 Question 3 The point 𝑥=−2 is Answer 2 Question 3 The point 𝑥=0 is Answer 3 Question 3 The point 𝑥=2 is Answer 4 Question 3 The point 𝑥=4 is Answer 5 Question 3

The function 𝑓(𝑥)=−3𝑥4+16𝑥3+24𝑥2−192𝑥+10 has critical points at some of the following 𝑥 -values. In each case identify whether the point is a critical point, and if so find out if it is a maximum, minimum or neither maximum nor minimum. The point 𝑥=−6 is Answer 1 Question 3 The point 𝑥=−2 is Answer 2 Question 3 The point 𝑥=0 is Answer 3 Question 3 The point 𝑥=2 is Answer 4 Question 3 The point 𝑥=4 is Answer 5 Question 3

The function 𝑓(𝑥,𝑦)=𝑥3−3𝑥−2𝑥𝑦−𝑦2−2𝑦+4 has two critical points. Find and classify them. Note: For non-integer numerical values, you must use at least 3 decimal places. Also, you must use a 'full stop' . and not a comma ',' for a decimal point. The left-most stationary point (the one with the lower value of 𝑥 ) is located at 𝑥= Answer 1 Question 2 , 𝑦= Answer 2 Question 2 , and is a Answer 3 Question 2 The right-most stationary point (the one with the higher value of 𝑥 ) is located at 𝑥= Answer 4 Question 2 , 𝑦= Answer 5 Question 2 , and is a Answer 6 Question 2

Find the 𝑥-values of both critical points of the following function. Then use the first derivative test to classify each of them as either a relative minimum, relative maximum, or neither.𝑦=−𝑥33−𝑥22+30𝑥−8 Find the first derivative.𝑦′=

Maximize the function 𝑓(𝑥,𝑦)=3𝑥𝑦−8𝑥2−4𝑥−5𝑦+12 subject to the constraint 3𝑥+2𝑦=5 Find the location of the maximum and its value. Enter non-integer numerical values as decimals to at least 3 decimal places. Note: you must use a . and not , for a decimal point. The maximum is located at 𝑥= Answer 1 Question 7 , 𝑦= Answer 2 Question 7 , and 𝑓(𝑥,𝑦)= Answer 3 Question 7

Use Lagrange multipliers to find the optimal value of the function 𝑓(𝑥,𝑦)=𝑥2+𝑦2+𝑥𝑦 subject to the constraint 𝑥−2𝑦=7 Find the location of the optimal point, the value of 𝜆 and the value of 𝑓(𝑥,𝑦) at this point. Enter non-integer numerical values as decimals to at least 3 decimal places. Note: you must use a . and not , for a decimal point. The optimal point is located at 𝑥= Answer 1 Question 5 , 𝑦= Answer 2 Question 5 , with 𝜆= Answer 3 Question 5 , and 𝑓(𝑥,𝑦)= Answer 4 Question 5