Advanced MathVery hardMinimum of a quadratic with a parameter
Advanced Math practice question
The function f is defined by f(x) = x² − 2kx + k² + 1, where k is a constant. What is the minimum value of f?
- A. 1Correct
- B. k² + 1
- C. −k²
- D. 0
Answer: A. 1
f(x) = (x − k)² + 1, and the square is at least 0, so the minimum is 1, reached at x = k.
Why the other answers are wrong
- B. k² + 1
- k² + 1 is f(0), which is the minimum only when k = 0.
- C. −k²
- The square cannot be negative.
- D. 0
- The +1 lifts the minimum above 0.
More Advanced Math questions
- Very hard
The function f(x) = x² + 6x + 5 is rewritten in the form f(x) = (x + a)² + b. What is the value of b? - Very hard
What is the solution to the equation 6/(x − 2) = 3? - Very hard
The polynomial p(x) = x³ − 4x has how many distinct real zeros? - Very hard
The polynomial p is defined by p(x) = x³ + ax² + bx − 12, where a and b are constants. If x − 2 is a factor of… - Very hard
The function f, defined by f(x) = 3000(1.331)^(x/12), gives the value, in dollars, of an investment x months a… - Very hard
In the xy-plane, the parabola y = x² − 4x + 7 and the line y = mx − 2, where m is a constant, intersect at exa…
Not affiliated with the College Board. SAT is a trademark of the College Board.