Question

Let $f^{\prime}(x)=\frac{192 x^3}{2+\sin ^4 \pi x}$ for all $x \in R$ with $f\left(\frac{1}{2}\right)=0$. If $m \leq \int\limits_{1 / 2}^1 f(x) d x \leq M$, then the possible values of $m$ and $M$ are

• A. $m=13, M=24$
• B. $m=\frac{1}{4}, M=\frac{1}{2}$
• C. $m=-11, M=0$
• D. $m=1, M=12$

Answer 1

Answer: (D)

$f^{\prime}(x)=\frac{192 x^3}{2+\sin ^4(\pi x)} \forall x \in R ; f\left(\frac{1}{2}\right)=0$
Now $64 x ^3 \leq f ^{\prime}( x ) \leq 96 x ^3 \forall x \in\left[\frac{1}{2}, 1\right]$
So $16 x^4-1 \leq f(x) \leq 24 x^4-\frac{3}{2} \forall x \in\left[\frac{1}{2}, 1\right]$
$ \frac{16}{5} \cdot \frac{31}{32}-\frac{1}{2} \leq \int_{y^2}^1 f(x) d x \leq \frac{24}{5} \cdot \frac{31}{32}-\frac{3}{4} $
$ \Rightarrow \frac{26}{10} \leq \int\limits_{1 / 2}^1 f(x) d x \leq \frac{78}{20} $ hence (D)
(1) is incorrect as $\int\limits_{1 / 2}^1 f(x) d x \leq \frac{78}{20}$
(2) is incorrect as $\int\limits_{1 / 2}^1 f(x) d x>\frac{26}{10}$
(3) is incorrect as $\int\limits_{y^2}^1 f(x) d x>0$