Find the intervals in which f(x) = (x + 2)e-x is increasing or de… - Vidyadip

Question

Find the intervals in which f(x) = (x + 2)e^-x is increasing or decreasing.

✓

Answer

f(x) = (x + 2)e^-x
f'(x) = -e^-x(x + 2) + e^-x
= -xe^-x - 2e^-x + e^-x
= -xe^-x - e^-x
= e^-x(-x - 1)
For f(x) to be increasing, we must have
f'(x) > 0
⇒ e^-x(-x - 1) > 0
⇒ -x - 1 > 0 $\big[\because\ \text{e}^{-\text{x}}>0,\forall\ \text{x}\in\text{R}\big]$
⇒ -x > 1
⇒ x < -1
$\Rightarrow\text{x}\in(-\infty,-1)$
So, f(x) is increasing on $(-\infty,-1).$
For f(x) to be decreasing, we must have
f'(x) < 0
⇒ e^-x(-x - 1) < 0
⇒ -x - 1 < 0 $\big[\because\ \text{e}^{-\text{x}}>0,\forall\ \text{x}\in\text{R}\big]$
⇒ -x < 1
⇒ x < -1
$\Rightarrow\text{x}\in(-1,\infty)$
So, f(x) is decreasing on $(-1,\infty).$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "4 Marks" from Increasing and Decreasing Functions→Increasing and Decreasing Functions — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Evalute the following integrals:
$\int\frac{\sin2\text{x}}{\text{a}\cos^2\text{x}+\text{b}\sin^2\text{x}}\text{dx}$

If $\text{x}=2\cos\theta-\cot2\theta$ and $\text{y}=2\sin\theta-\sin2\theta,$ prove that $\frac{\text{dy}}{\text{dx}}=\tan\big(\frac{3\theta}{2}\big)$

Differentiate the following functions from first principles:
$\text{e}^{\sqrt{\cot\text{x}}}$

Find the distance between the point (7, 2, 4) and the plane determined by the points A(2, 5, -3), B(-2, -3, 5) nad C(5, 3, -3).

Evalute the following integrals:
$\int\frac{-\sin\text{x}+2\cos\text{x}}{2\sin\text{x}+\cos\text{x}}\text{dx}$

Evaluate the following integrals as limit of sum:
$\int\limits^2_{0}\text{e}^{\text{x}}\text{ dx}$

Solve the following differential equation
$\text{C}(\text{x})=2+0.15\text{x},\text{C}(0)=100$

Differentiate the following with respect to x:
$\cos^{-1}\Big(\frac{1-\text{x}}{1+\text{x}}\Big)$

Verify mean value theorem for the function:
$\text{f(x)}=\sqrt{25-\text{x}^2}\text{ in }[1,5].$

A company manufactures two types of novelty souvenirs made of plywood. Souvenirs of type A require 5 minutes each for cutting and 10 minutes each for assembling. Souvenirs of type B require 8 minutes each for cutting and 8 minutes each for assembling. There are 3 hours 20 minutes available for cutting and 4 hours for assembling. The profit is Rs 5 each for type A and Rs 6 each for type B souvenirs. How many souvenirs of each type should the company manufacture in order to maximise the profit?