Function negative for all values of x

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The function 3x^2 - 12x + m can only be negative for all values of x if it does not intersect the x-axis. The discriminant condition b^2 - 4ac < 0 leads to the conclusion that m must be greater than 12. However, substituting values like m = 13 shows that the function can still yield positive results for certain x values. The graph of the function is a parabola that opens upwards, indicating that it cannot remain below the x-axis for all x. Therefore, no value of m exists that keeps the function negative for all x.
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Homework Statement


Find the range of values of m for which the function 3x^2 -12x + m is negative for all values of x.

Homework Equations


Is it possible for the function to be negative ? If so, how ?

The Attempt at a Solution


I tried using b^2 - 4ac < 0 and the result is m > 12 . However when i tried substituting *13* as the value of m and *1* as the value of x, the result is a positive number.
 
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I don't think that any such value of m exists. Just picture this intuitively. What does the graph look like? Is it shaped like a smiley expression or sad expression? Moreover, even if the discriminant is less than zero it just means that the graph doesn't cut the x-axis. It doesn't mean that it is below the x-axis.
 
Defennder is correct: No such value of m exists. Complete the square, or just see that 3x^2 - 12x itself can be made as large as required to cancel out any value of m.
 

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