Which of the following statements is FALSE?
A - HSC - SSCE Mathematics Extension 2 - Question 5 - 2021 - Paper 1
Question 5
Which of the following statements is FALSE?
A. $\forall a, b \in \mathbb{R}, \quad a < b \implies a^3 < b^3$
B. $\forall a, b \in \mathbb{R}, \quad a < b \impli... show full transcript
Worked Solution & Example Answer:Which of the following statements is FALSE?
A - HSC - SSCE Mathematics Extension 2 - Question 5 - 2021 - Paper 1
Step 1
A. $\forall a, b \in \mathbb{R}, \quad a < b \implies a^3 < b^3$
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Answer
This statement is true because the function f(x)=x3 is an increasing function for all real numbers.
Step 2
B. $\forall a, b \in \mathbb{R}, \quad a < b \implies e^a > e^b$
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Answer
This statement is false. The exponential function ex is increasing, implying that if a<b, then ea<eb, not greater.
Step 3
C. $\forall a, b \in (0, +\infty), \quad a < b \implies \ln a < \ln b$
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Answer
This statement is true because the logarithm function is increasing for positive arguments.
Step 4
D. $\forall a, b \in \mathbb{R}, \text{ with } a, b \neq 0, \quad a < b \implies \frac{1}{a} > \frac{1}{b}$
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Answer
This statement is true, provided that a and b are both positive. For negative values, the inequality reverses.
