Question

1. $f(x) = -x^3 + 4x^2 - 5$

a) graph
b) graph
c) graph
d) graph
options: a, c, d (selected), b

Explanation:

Step1: Analyze Leading Term

The function is \( f(x) = -x^3 + 4x^2 - 5 \). The leading term is \( -x^3 \), so as \( x \to +\infty \), \( f(x) \to -\infty \), and as \( x \to -\infty \), \( f(x) \to +\infty \). This eliminates option A (which goes to \( +\infty \) as \( x \to +\infty \)).

Step2: Find y - intercept

Set \( x = 0 \): \( f(0) = -0 + 0 - 5 = -5 \). So the graph passes through \( (0, -5) \). Check options:

• Option B: At \( x = 0 \), the y - value seems positive, eliminate.
• Option C: At \( x = 0 \), the y - value seems around 0, eliminate.
• Option D: At \( x = 0 \), the y - value is - 5 (matches), and the end - behavior (as \( x\to+\infty \), \( y\to-\infty \); as \( x\to-\infty \), \( y\to+\infty \)) also matches.

Answer:

D