Suppose $f(x)$ is a differentiable function such that
$$\frac{f(a)+f(b)}{2} \geq f\left( \frac{a+b}{2} \right)$$ for all $a$ and $b$ - HSC - SSCE Mathematics Extension 2 - Question 10 - 2017 - Paper 1
Question 10
Suppose $f(x)$ is a differentiable function such that
$$\frac{f(a)+f(b)}{2} \geq f\left( \frac{a+b}{2} \right)$$ for all $a$ and $b$.
Which statement is always true... show full transcript
Worked Solution & Example Answer:Suppose $f(x)$ is a differentiable function such that
$$\frac{f(a)+f(b)}{2} \geq f\left( \frac{a+b}{2} \right)$$ for all $a$ and $b$ - HSC - SSCE Mathematics Extension 2 - Question 10 - 2017 - Paper 1
Step 1
A. $\int_0^1 f(x)dx \geq \frac{f(0)+f(1)}{2}$
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Answer
This statement cannot be guaranteed to always be true, as it depends on the specific function f(x).
Step 2
B. $\int_0^1 f(x)dx \leq \frac{f(0)+f(1)}{2}$
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Answer
This statement is indeed always true based on the given condition. The inequality suggests the function is concave up, which leads to this conclusion through Jensen's inequality.
Step 3
C. $f'\left( \frac{1}{2} \right) \geq 0$
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Answer
This statement cannot be established as a certainty without additional information about the function f(x).
Step 4
D. $f'\left( \frac{1}{2} \right) \leq 0$
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Answer
Similar to statement C, this cannot be determined to always be true without further context.
