SATHEE: Application of Derivatives 4 Question 32

Application of Derivatives 4 Question 32

####34. If $f (x) = 2 - e^{x - 1}, 1 < x \leq 2$ and $g (x) = \int_{0}^{x} f (t) d t$ ,

$x - e, 2 < x \leq 3$

$x \in [1, 3]$ , then

(2006, 3M)

(a) $g (x)$ has local maxima at $x = 1 + \log_{e} 2$ and local minima at $x = e$

(b) $f (x)$ has local maxima at $x = 1$ and local minima at $x = 2$

(d) $f (x)$ has no local maxima

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Answer:

Correct Answer: 34. (b, d)

Solution:

Given,

$g^{''} (1 + \log_{e} 2) = - e^{\log_{e} 2} < 0, g (x)$ has a local maximum.

Also, at $x = e$ ,

$g^{''} (e) = 1 > 0, g (x)$ has a local minima.

$∵ f (x)$ is discontinuous at $x = 1$ , then we get local maxima at $x = 1$ and local minima at $x = 2$ .

Hence, (a) and (b) are correct answers.

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Application of Derivatives 4 Question 32

Answer:

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