Multiple choice

Let $f(x)$ be a quadratic expression which is positive $\forall x\in R$ . Then, the value of $10[f(x)+f(-x)]$ is

  1. $>0$
  2. $\geq 0$
  3. $<0$
  4. $\leq 0$
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A Correct answer
Explanation

For a quadratic f(x) = ax^2 + bx + c to be positive for all x, a must be > 0 and the discriminant must be < 0. f(x) + f(-x) = (ax^2 + bx + c) + (ax^2 - bx + c) = 2ax^2 + 2c. Since a > 0 and c > 0 (as f(0) > 0), this expression is always positive.

AI explanation

Because f(x) is a quadratic expression that is positive for all real x its leading coefficient must be strictly positive. Expanding 10[f(x) + f(-x)] eliminates all odd powers of x leaving only twice the even powers multiplied by 10. Since the leading term remains strictly positive and f(x) is positive everywhere the value of 10[f(x) + f(-x)] is strictly greater than 0. The result is > 0.

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