Multiple choice

Find the range of the quadratic expression for real $x$

  1. $3x^{2}+2x+11$
  2. $\sin^{2} x+\sin x+2$
  3. $4^{x}-2^{x}+1$
  4. $4^{x}+2^{2}+1$
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A Correct answer
AI explanation

The question asks to identify a valid range of a quadratic expression from the given options. Option A, the expression 3x^2 + 2x + 11, can be rewritten by completing the square as 3(x + 1/3)^2 + 32/3. Because (x + 1/3)^2 is always greater than or equal to zero for real x, the minimum value of the expression is 32/3. Therefore, the range of this expression is [32/3, infinity), demonstrating a correct range of a quadratic expression for real x.

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