Problem 41 - A Graphical Approach to Precalculus with Limits

Question

Answer true or false to each statement. Then support your answer by graphing. The function \(f(x)=-3 x^{4}+5\) has two real zeros.

Step-by-Step Solution

Answer

False, the function has two real zeros.

Step 1: Understand the Given Function

The function given is \( f(x) = -3x^4 + 5 \). This is a polynomial function of degree 4, specifically a quartic function.

Step 2: Analyze the End Behavior

Since the leading term is \(-3x^4\), a negative coefficient indicates that as \(x\) approaches \(±∞\), the function \(f(x)\) will approach \(−∞\). This suggests that the graph opens downwards.

Step 3: Evaluate the Function at Critical Points

Check the values of \(f(x)\) at potential critical points to see if it crosses the x-axis. Setting \(f(x) = 0\) gives \(-3x^4 + 5 = 0\). Solve for \(x\), we get \(-3x^4 = -5\) or \(x^4 = \frac{5}{3}\). The solutions are \(x = \pm (\frac{5}{3})^{1/4}\).

Step 4: Graph the Function

The function will cross the x-axis at two points at \(x = \pm (\frac{5}{3})^{1/4}\). The graph will have a maximum point as it opens downwards, providing evidence of two real zeros.

Step 5: Conclusion from the Graph

By drawing the graph of \( f(x) = -3x^4 + 5 \), it is evident that the function does indeed intersect the x-axis at two points, confirming the presence of two real zeros.

Key Concepts

Real Zeros

Real zeros of a polynomial function are the values of \(x\) for which the function equals zero. In other words, they are the points where the graph of the function crosses the x-axis. Identifying real zeros is important as it helps understand the root structure of the function and provides insight into the graph's behavior.
To find real zeros of the function \(f(x) = -3x^4 + 5\), we set the function equal to zero: \(-3x^4 + 5 = 0\). Solving for \(x\), we obtain \(-3x^4 = -5\) or \(x^4 = \frac{5}{3}\). This implies that \(x = \pm \left(\frac{5}{3}\right)^{1/4}\). These solutions show that \(f(x)\) has two real zeros. These are the points where the function intersects the x-axis. Since these values of \(x\) yield \(f(x) = 0\), they are confirmed as the real zeros of \(f(x)\).

End Behavior

The end behavior of a polynomial function describes how the value of \(f(x)\) behaves as \(x\) becomes very large (positively or negatively). For \(f(x) = -3x^4 + 5\), the end behavior can be determined by the leading term, \(-3x^4\). Leading terms are crucial because they dominate the behavior of the polynomial as \(x\) approaches infinity.

With a quartic leading term and a negative coefficient \(-3\), as \(x\) approaches \(\pm \infty\), the term \(x^4\) (being positive) is multiplied by a negative, sending \(f(x)\) to \(-\infty\). This means that both ends of the graph move downwards, indicating the graph opens downwards on both sides.
Understanding this behavior is helpful in predicting the general shape of the graph without needing precise calculations or graph plotting at every point. It allows quick comprehension of the orientation and trajectory of the polynomial function's graph as it extends away from the origin.

Quartic Function

A quartic function is a polynomial of degree four, which means the highest power of \(x\) in the function is four. An example is \(f(x) = -3x^4 + 5\). The general form of a quartic function is \(ax^4 + bx^3 + cx^2 + dx + e\), where \(a, b, c, d, \) and \(e\) are constants, and \(a eq 0\).
Quartic functions can manifest various graph shapes depending on their coefficients, particularly those affecting the end behavior. Because the degree of the polynomial is even, the arms of the graph (end behavior) will either both point upwards or downwards. For the given function, \(-3x^4 + 5\), the negative leading coefficient means it opens downwards.
The number of possible real zeros for a quartic function depends on its coefficients, ranging from 0 to 4 real zeros. For \(f(x) = -3x^4 + 5\), we found two real zeros by solving for \(x\) when the function equals zero. Analyzing quartic functions can seem complex, but understanding the degree and leading coefficient immensely aids in predicting the function's overall shape and behavior.