0
$\begingroup$

Let $f \in \mathcal{C}[a,b]$. Assume that $min_{x \in [a,b]} f(x)=m>0$ and let $M=max_{ x \in [a,b]}f(x)$. Which of the following inequalities are true?

$(A)$ $\frac{1}{M} \int\limits_a^b f(x) \ dx + m \int\limits_a^b \frac{1}{f(x)} \ dx \geq 2 \sqrt{\frac{m}{M}}(b-a)$

$(B)$ $\int\limits_a^b f(x) \ dx \int\limits_a^b \frac{1}{f(x)} \ dx \geq (b-a)^2$

$(C)$ $ \int\limits_a^b f(x) \ dx \int\limits_a^b \frac{1}{f(x)} \ dx \leq (b-a)^2$

So according to answer at the back $A \& B$ are the correct options. $B$ is clear to me you replace $f$ and $1/f$ on the $LHS$ by its appropriate bounds to attain the inequality. I am not sure how to attack $A$. Any help?

  • 0
    Is it a homework? This same question has already appeared here at least two more time within the last hour. – 2017-01-19
  • 0
    You only have to use $f(x) \leq M, f(x) \geq m, 1/f(x) \leq 1/m, 1/f(x) \geq 1/M$. – 2017-01-19
  • 0
    @zipirovich No. I am working through previous years question papers from the Notional Board of Mathematics (India). – 2017-01-19
  • 0
    Apparently, tonight's the night for working thru last year's NBMI. See http://math.stackexchange.com/questions/2104065/nbhm-question-2013-riemann-integration-related-question – 2017-01-19

0 Answers 0