SATHEE: Indefinite Integration 1 Question 46

Indefinite Integration 1 Question 46

47. Let $f : R \to (0, 1)$ be a continuous function. Then, which of the following function(s) has (have) the value zero at some point in the interval $(0, 1)$ ?

(2017 Adv.)

(a) $e^{x} - \int_{0}^{x} f (t) \sin t d t$

(b) $f (x) + \int_{0}^{\frac{π}{2}} f (t) \sin t d t$

(c) $x - \int_{0} f (t) \cos t d t$

(d) $x^{9} - f (x)$

Show Answer

Solution:

$∵ e^{x} \in (1, e)$ in $(0, 1)$ and $\int_{0}^{x} f (t) \sin t d t \in (0, 1)$ in $(0, 1)$

$∴ e^{x} - \int_{0}^{x} f (t) \sin t d t$ cannot be zero.

So, option (a) is incorrect. (b) $f (x) + \int_{0}^{\frac{π}{2}} f (t) \sin t d t$ always positive

$∴$ Option (b) is incorrect.

(c) Let $h (x) = x - \int_{0}^{2} f (t) \cos t d t$ ,

$\begin{aligned} h (0) = - \int_{0}^{\frac{π}{2}} f (t) \cos t d t < 0 \\ h (1) = 1 - \int_{0}^{\frac{π}{2} - 1} f (t) \cos t d t > 0 \end{aligned}$

$∴$ Option (c) is correct.

(d) Let $g (x) = x^{9} - f (x)$

$g (0) = - f (0) < 0$

$g (1) = 1 - f (1) > 0$

$∴$ Option (d) is correct.

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