Question 7:
(2 marks)
The function \( f(x)=x^{4}-6 x^{2}-8 \) is concaving up over the interval.
a) \( (-1,1) \)
b) \( (-\infty,-1) \mathrm{U}(1, \infty) \)
c) \( (-\infty,-1) \mathrm{U}(-1,1) \mathrm{U}(1, \infty) \)
d) none of them

Answer

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Answer

\(f''(x) \ge 0 \Rightarrow x^2 \ge 1 \Rightarrow x \in (-\infty,-1) \cup (1, \infty)\)

Steps

Step 1 ：\(f'(x) = 4x^3 - 12x\)

Step 2 ：\(f''(x) = 12x^2 - 12\)

Step 3 ：\(f''(x) \ge 0 \Rightarrow x^2 \ge 1 \Rightarrow x \in (-\infty,-1) \cup (1, \infty)\)