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Consider the function f(x)=-(1)/(3)(x-2)^(2)+1. Algebraically determine which of the following are the zeros for f(x). Select all correct answers below. (A) 0 (B) 3+sqrt2 (C) 4 (D) 2+sqrt3 (E) 3-sqrt2 (F) 2-sqrt3

Consider the function f(x)=−13(x−2)2+1 f(x)=-\frac{1}{3}(x-2)^{2}+1 . Algebraically determine which of the following are the zeros for f(x) f(x) . Select all correct answers below.\newline(A) 00\newline(B) 3+2 3+\sqrt{2} \newline(C) 44\newline(D) 2+3 2+\sqrt{3} \newline(E) 3−2 3-\sqrt{2} \newline(F) 2−3 2-\sqrt{3}

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Q. Consider the function f(x)=−13(x−2)2+1 f(x)=-\frac{1}{3}(x-2)^{2}+1 . Algebraically determine which of the following are the zeros for f(x) f(x) . Select all correct answers below.\newline(A) 00\newline(B) 3+2 3+\sqrt{2} \newline(C) 44\newline(D) 2+3 2+\sqrt{3} \newline(E) 3−2 3-\sqrt{2} \newline(F) 2−3 2-\sqrt{3}
  1. Define Zero of Function: Understand what a zero of a function is.\newlineA zero of a function is a value of xx for which the function equals zero. To find the zeros of f(x)f(x), we need to solve the equation f(x)=0f(x) = 0.
  2. Set Function Equal: Set the function equal to 00 and solve for xx. We have f(x)=−(13)(x−2)2+1f(x) = -(\frac{1}{3})(x - 2)^2 + 1. To find the zeros, we set f(x)f(x) to 00: 0=−(13)(x−2)2+10 = -(\frac{1}{3})(x - 2)^2 + 1
  3. Isolate Squared Term: Isolate the squared term.\newlineAdd (13)(x−2)2(\frac{1}{3})(x - 2)^2 to both sides to isolate the squared term:\newline(13)(x−2)2=1(\frac{1}{3})(x - 2)^2 = 1
  4. Eliminate Fraction: Multiply both sides by 33 to eliminate the fraction.3×(13)(x−2)2=3×13 \times \left(\frac{1}{3}\right)(x - 2)^2 = 3 \times 1(x−2)2=3(x - 2)^2 = 3
  5. Take Square Root: Take the square root of both sides.\newline(x−2)2=±3\sqrt{(x - 2)^2} = \pm\sqrt{3}\newlinex−2=±3x - 2 = \pm\sqrt{3}
  6. Solve for x: Solve for x by adding 22 to both sides.\newlinex=2±3x = 2 \pm \sqrt{3}
  7. Identify Solutions: Identify the two solutions.\newlineThe two solutions are:\newlinex=2+3x = 2 + \sqrt{3}\newlinex=2−3x = 2 - \sqrt{3}
  8. Check Answer Choices: Check the answer choices against the solutions.\newlineThe correct answer choices that match our solutions are:\newlineD) 2+32 + \sqrt{3}\newlineF) 2−32 - \sqrt{3}

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