IB Mathematics SL 5.2 Increasing and decreasing functions AA SL Paper 2- Exam Style Questions- New Syllabus

(a)
The function \( f \) is decreasing when \( f'(x) < 0 \).
Solve \( 4 + 2x – 3e^x < 0 \).
Using a GDC to find the roots of \( 4 + 2x – 3e^x = 0 \):
\( x \approx -1.73554 \) and \( x \approx 0.517999 \).
Testing intervals or using the graph of \( f'(x) \):
\( f'(x) < 0 \) when \( x < -1.73554 \) or \( x > 0.517999 \).
\( \boxed{x \leq -1.74 \text{ and } x \geq 0.518} \) (or with strict inequalities).

(b)
The graph of \( f \) is concave-up when \( f”(x) > 0 \).
Find \( f”(x) \) by differentiating \( f'(x) \):
\( f”(x) = 2 – 3e^x \).
Set \( f”(x) > 0 \):
\( 2 – 3e^x > 0 \)
\( 3e^x < 2 \)
\( e^x < \frac{2}{3} \)
\( x < \ln\left(\frac{2}{3}\right) \)
\( x < -0.405465… \)
\( \boxed{x < \ln\left(\frac{2}{3}\right)} \) or \( x < -0.405 \).