Advertisements
Advertisements
Question
Find the intervals in which f(x) = (x + 2) e−x is increasing or decreasing ?
Advertisements
Solution
\[f\left( x \right) = \left( x + 2 \right) e^{- x} \]
\[f'\left( x \right) = - e^{- x} \left( x + 2 \right) + e^{- x} \]
\[ = - x e^{- x} - 2 e^{- x} + e^{- x} \]
\[ = - x e^{- x} - e^{- x} \]
\[ = e^{- x} \left( - x - 1 \right)\]
\[\text { For f(x) to be increasing, we must have }\]
\[f'\left( x \right) > 0\]
\[ \Rightarrow e^{- x} \left( - x - 1 \right) > 0\]
\[ \Rightarrow - x - 1 > 0 \left[ \because e^{- x} > 0, \forall x \in R \right]\]
\[ \Rightarrow - x > 1\]
\[ \Rightarrow x < - 1\]
\[ \Rightarrow x \in \left( - \infty , - 1 \right)\]
\[\text { So, f(x) is increasing on} \left( - \infty , - 1 \right) . \]
\[\text { For f(x) to be decreasing, we must have }\]
\[f'\left( x \right) < 0\]
\[ \Rightarrow e^{- x} \left( - x - 1 \right) < 0\]
\[ \Rightarrow - x - 1 < 0 \left[ \because e^{- x} > 0, \forall x \in R \right]\]
\[ \Rightarrow - x < 1\]
\[ \Rightarrow x > - 1\]
\[ \Rightarrow x \in \left( - 1, \infty \right)\]
\[\text { So, f(x) is decreasing on }\left( - 1, \infty \right).\]
APPEARS IN
RELATED QUESTIONS
Prove that the function f given by f(x) = log cos x is strictly decreasing on `(0, pi/2)` and strictly increasing on `((3pi)/2, 2pi).`
Prove that the function f(x) = loga x is increasing on (0, ∞) if a > 1 and decreasing on (0, ∞), if 0 < a < 1 ?
Prove that f(x) = ax + b, where a, b are constants and a < 0 is a decreasing function on R ?
Without using the derivative, show that the function f (x) = | x | is.
(a) strictly increasing in (0, ∞)
(b) strictly decreasing in (−∞, 0) .
Find the interval in which the following function are increasing or decreasing f(x) = 10 − 6x − 2x2 ?
Find the interval in which the following function are increasing or decreasing f(x) = 6 − 9x − x2 ?
Find the interval in which the following function are increasing or decreasing f(x) = (x − 1) (x − 2)2 ?
Find the interval in which the following function are increasing or decreasing f(x) = x4 − 4x ?
Show that the function x2 − x + 1 is neither increasing nor decreasing on (0, 1) ?
Prove that the function f given by f(x) = log cos x is strictly increasing on (−π/2, 0) and strictly decreasing on (0, π/2) ?
Find the interval in which f(x) is increasing or decreasing f(x) = x|x|, x \[\in\] R ?
Find the interval in which f(x) is increasing or decreasing f(x) = sinx(1 + cosx), 0 < x < \[\frac{\pi}{2}\] ?
Find 'a' for which f(x) = a (x + sin x) + a is increasing on R ?
The interval of increase of the function f(x) = x − ex + tan (2π/7) is
The function f(x) = xx decreases on the interval
Let ϕ(x) = f(x) + f(2a − x) and f"(x) > 0 for all x ∈ [0, a]. Then, ϕ (x)
Show that the function f given by f(x) = tan–1 (sin x + cos x) is decreasing for all \[x \in \left( \frac{\pi}{4}, \frac{\pi}{2} \right) .\]
Find the intervals in which function f given by f(x) = 4x3 - 6x2 - 72x + 30 is (a) strictly increasing, (b) strictly decresing .
Show that y = `log (1 + x) – (2x)/(2 + x), x > - 1` is an increasing function on its domain.
Show that the function f(x) = x3 + 10x + 7 for x ∈ R is strictly increasing
Test whether the function f(x) = x3 + 6x2 + 12x − 5 is increasing or decreasing for all x ∈ R
Find the values of x for which the function f(x) = 2x3 – 6x2 + 6x + 24 is strictly increasing
The total cost function for production of articles is given as C = 100 + 600x – 3x2, then the values of x for which the total cost is decreasing is ______
For every value of x, the function f(x) = `1/"a"^x`, a > 0 is ______.
The area of the square increases at the rate of 0.5 cm2/sec. The rate at which its perimeter is increasing when the side of the square is 10 cm long is ______.
Determine for which values of x, the function y = `x^4 – (4x^3)/3` is increasing and for which values, it is decreasing.
Let the f : R → R be defined by f (x) = 2x + cosx, then f : ______.
The function f(x) = tanx – x ______.
Let `"f (x) = x – cos x, x" in "R"`, then f is ____________.
Let f (x) = tan x – 4x, then in the interval `[- pi/3, pi/3], "f"("x")` is ____________.
The function `"f"("x") = "log" (1 + "x") - (2"x")/(2 + "x")` is increasing on ____________.
The function `"f"("x") = "x"/"logx"` increases on the interval
Function given by f(x) = sin x is strictly increasing in.
State whether the following statement is true or false.
If f'(x) > 0 for all x ∈ (a, b) then f(x) is decreasing function in the interval (a, b).
Let f : R `rightarrow` R be a positive increasing function with `lim_(x rightarrow ∞) (f(3x))/(f(x))` = 1 then `lim_(x rightarrow ∞) (f(2x))/(f(x))` = ______.
