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CAT Maxima and Minima questions with solutions

Optimisation of expressions. 11 questions from our 2016–2021 CAT pattern papers. Each has the answer and a full solution.

CAT 2021

If x, y and z are positive real numbers with x + y + z = 1, what is the minimum value of 1/x + 4/y + 9/z?CAT 2021 Slot 1 · hard
Real numbers a, b and c satisfy a + b + c = 6 and ab + bc + ca = 9. What is the maximum possible value of c?CAT 2021 Slot 3 · hard

CAT 2020

If x and y are positive real numbers with x + y = 12, the maximum possible value of x²y isCAT 2020 Slot 1 · hard
Real numbers x and y satisfy 3x + 4y = 25. What is the minimum possible value of x² + y²?CAT 2020 Slot 2 · hard

CAT 2019

X and y are positive real numbers with x + y = 10. What is the minimum possible value of x² + 4y²?CAT 2019 Slot 1 · hard
If x and y are real numbers with x + y = 10, what is the minimum possible value of x² + xy + y²?CAT 2019 Slot 2 · medium

CAT 2018

If x and y are real numbers with x + y = 10, what is the minimum value of x² + y² + xy?CAT 2018 Slot 2 · medium

CAT 2017

What is the minimum value of f(x) = max(2x + 1, 7 − x) over all real numbers x?CAT 2017 Slot 1 · medium
X and y are positive real numbers with x + y = 10. What is the maximum possible value of x²y?CAT 2017 Slot 2 · hard

CAT 2016

For x > −2, the minimum value of (x² + 4x + 13)/(x + 2) isCAT 2016 Slot 1 · hard
If x and y are positive real numbers with x + y = 12, what is the maximum value of x²y?CAT 2016 Slot 2 · hard