$-3 x^4+5$ extends in Quadrant 3 and Quadrant 4. (True/False)
Answer (1)
The function is f ( x ) = − 3 x 4 + 5 .
Determine if the function takes on negative values.
The range of the function is ( − ∞ , 5 ] .
Since the range includes negative values, the statement is true. T r u e
Explanation
Understanding the Problem We are given the function f ( x ) = − 3 x 4 + 5 and asked whether it extends in Quadrant 3 and Quadrant 4. Recall that Quadrants 3 and 4 are the quadrants where the y -values are negative. Thus, we need to determine if the function f ( x ) takes on negative values for some x .
Analyzing the Function To determine if f ( x ) takes on negative values, we can analyze the function. The term x 4 is always non-negative, so x 4 ≥ 0 . Multiplying by − 3 , we have − 3 x 4 ≤ 0 . Adding 5, we get − 3 x 4 + 5 ≤ 5 . Thus, the function is always less than or equal to 5.
Finding the Range Now, let's find the range of the function. The maximum value occurs when x = 0 , and f ( 0 ) = − 3 ( 0 ) 4 + 5 = 5 . As x becomes large (either positive or negative), the term − 3 x 4 becomes a large negative number, so f ( x ) approaches − ∞ . Therefore, the range of the function is ( − ∞ , 5 ] .
Determining if the Function Extends in Quadrants 3 and 4 Since the range of the function is ( − ∞ , 5 ] , it includes negative values. This means that the function extends in Quadrants 3 and 4.
Conclusion Therefore, the statement ' − 3 x 4 + 5 extends in Quadrant 3 and Quadrant 4' is true.
Examples
Understanding the behavior of functions like − 3 x 4 + 5 is crucial in various real-world applications. For instance, in physics, this could represent the potential energy of a system, where the negative values indicate a stable state. In engineering, it might describe the deflection of a beam under load, where negative values represent downward displacement. Recognizing that the function extends into negative values helps engineers design structures that can withstand these deflections without failure. This knowledge ensures safer and more reliable designs.
