Find the intervals of monotonicity for the following functions & represent your solution set on the number line $f (x) = 2 . e^{x^{2} - 4 x}$

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- I in (3,∞) & D in (−∞,3)

- I in (2,∞) & D in (−∞,2)

- I in (1,∞) & D in (−∞,1)

- I in (4,∞) & D in (−∞,4)

answer is B.

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Detailed Solution

We have the function f(x)=2.

e^{x^{2} - 4 x}

.
Differentiating above function and applying chain rule, we get,
f′(x)=2(2x−4).

e^{x^{2} - 4 x}

For Increasing: f′(x)> 0 i.e.
f′(x)=2(2x−4).

e^{x^{2} - 4 x}

> 0
∵

e^{x^{2} - 4 x}

> 0 for all values of x . So for f′(x) > 0 we have,
2(2x−4) > 0
⇒4(x−2) > 0
⇒(x−2) > 0 (Since, 4 is constant)
⇒x>2
∴ For all values of x > 2 , f(x) is increasing
i.e. f(x) is increasing in the interval (2,∞)
For Decreasing: f′(x) < 0
f′(x)=2(2x−4).

e^{x^{2} - 4 x}

< 0
∵

e^{x^{2} - 4 x}

> 0 for all values of x . So for f′(x) < 0 we have,
2(2x−4) < 0
⇒4(x−2) < 0
⇒(x−2) < 0 (Since, 4 is constant)
⇒x < 2
∴ For all values of x < 2 , f(x) is decreasing
i.e. f(x) is decreasing in the interval (−∞,2)
So, f(x) is increasing in the interval (2,∞) and decreasing in the interval (−∞,2)
So, the correct answer is “Option 2”.

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