Advertisements
Advertisements
Question
Find the interval/s in which the function f : R `rightarrow` R defined by f(x) = xex, is increasing.
Advertisements
Solution
f(x) = xex `\implies` f'(x) = ex (x + 1)
When x ∈ [–1, ∞), (x + 1) ≥ 0 and ex > 0
`\implies` f'(x) ≥ 0
∴ f(x) increases in this interval.
or, we can write f(x) = xex `\implies` f'(x) = ex (x + 1)
For f(x) to be increasing, we have f'(x) = ex (x + 1) ≥ 0 `\implies` x ≥ –1 as ex > 0, ∀ x ∈ R
Hence, the required interval where f(x) increases is [–1, ∞).
APPEARS IN
RELATED QUESTIONS
Find the values of x for `y = [x(x - 2)]^2` is an increasing function.
Prove that the logarithmic function is strictly increasing on (0, ∞).
Find the interval in which the following function are increasing or decreasing f(x) = 5x3 − 15x2 − 120x + 3 ?
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \frac{x^4}{4} + \frac{2}{3} x^3 - \frac{5}{2} x^2 - 6x + 7\] ?
Show that the function x2 − x + 1 is neither increasing nor decreasing on (0, 1) ?
Determine whether f(x) = −x/2 + sin x is increasing or decreasing on (−π/3, π/3) ?
Prove that the following function is increasing on R f \[f\left( x \right) = 4 x^3 - 18 x^2 + 27x - 27\] ?
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) ?
Write the set of values of 'a' for which f(x) = loga x is increasing in its domain ?
Find 'a' for which f(x) = a (x + sin x) + a is increasing on R ?
The function f(x) = x2 e−x is monotonic increasing when
Function f(x) = x3 − 27x + 5 is monotonically increasing when ______.
Function f(x) = 2x3 − 9x2 + 12x + 29 is monotonically decreasing when
Function f(x) = | x | − | x − 1 | is monotonically increasing when
If the function f(x) = cos |x| − 2ax + b increases along the entire number scale, then
Function f(x) = loga x is increasing on R, if
The price P for demand D is given as P = 183 + 120 D – 3D2.
Find D for which the price is increasing.
Find MPC ( Marginal propensity to Consume ) and APC ( Average Propensity to Consume ) if the expenditure Ec of a person with income I is given as Ec = ( 0.0003 ) I2 + ( 0.075 ) I when I = 1000.
If the demand function is D = 50 - 3p - p2, find the elasticity of demand at (a) p = 5 (b) p = 2 , Interpret your result.
Test whether the following functions are increasing or decreasing: f(x) = `x-(1)/x`, x ∈ R, x ≠ 0.
Find the values of x for which the following functions are strictly decreasing : f(x) = x3 – 9x2 + 24x + 12
A circular pIate is contracting at the uniform rate of 5cm/sec. The rate at which the perimeter is decreasing when the radius of the circle is 10 cm Jong is
Given P(x) = x4 + ax3 + bx2 + cx + d such that x = 0 is the only real root of P'(x) = 0. If P(-1) < P(1), then in the interval [-1, 1] ______
The function f(x) = sin x + 2x is ______
If f(x) = `x^(3/2) (3x - 10)`, x ≥ 0, then f(x) is increasing in ______.
Prove that the function f(x) = tanx – 4x is strictly decreasing on `((-pi)/3, pi/3)`
The function f(x) = 4 sin3x – 6 sin2x + 12 sinx + 100 is strictly ______.
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).
If f(x) = `x/(x^2 + 1)` is increasing function then the value of x lies in ______.
