Question

Which of the two functions below has the largest maximum y value? f(x) = -3x^4 - 14, g(x) = -x^3 + 2

Answer 1

To find which function has the largest maximum \(y\) value, we need to analyze both functions \(f(x)\) and \(g(x)\).

1. **Analyzing \(f(x) = -3x^4 - 14\)**:
- The term \(-3x^4\) is a quartic term that opens downward (since the coefficient of \(x^4\) is negative). This means \(f(x)\) has a maximum point.
- To find the maximum value, we can evaluate \(f(x)\) at \(x = 0\):
\[
f(0) = -3(0)^4 - 14 = -14.
\]
- Since the leading term dominates as \(x\) moves away from zero (either positively or negatively), the maximum value of \(f(x)\) is \(-14\).

2. **Analyzing \(g(x) = -x^3 + 2\)**:
- The term \(-x^3\) is a cubic term that also opens downward but does not have a global maximum as it goes to \(+\infty\) as \(x\) goes to negative infinity and \(-\infty\) as \(x\) goes to positive infinity.
- To find the local maximum, we can find its derivative and set it to zero:
\[
g'(x) = -3x^2.
\]
Setting \(g'(x) = 0\) gives:
\[
-3x^2 = 0 \implies x = 0.
\]
- We can evaluate \(g(x)\) at \(x = 0\):
\[
g(0) = -0^3 + 2 = 2.
\]
- As \(g(x)\) increases to \(2\) when \(x\) is at \(0\), it does not attain values higher than \(2\) at any other real \(x\).

3. **Comparing the maximum values**:
- Maximum value of \(f(x)\) is \(-14\).
- Maximum value of \(g(x)\) is \(2\).

Since \(2 > -14\), we conclude that \(g(x)\) has the largest maximum \(y\) value.

Thus, the function with the largest maximum \(y\) value is \(g(x)\).