Advertisements
Advertisements
Question
Choose the correct alternative.
The function f(x) = x3 - 3x2 + 3x - 100, x ∈ R is
Options
increasing for all x ∈ R, x ≠ 1
decreasing
neither, increasing nor decreasing
decreasing for all x ∈ R, x ≠ 1
Advertisements
Solution
increasing for all x ∈ R, x ≠ 1
Explanation:
f(x) = x3 - 3x2 + 3x - 100
Differentiating w.r.t. x, we get
f'(x) = 3x2 - 6x + 3
= 3(x2 - 2x + 1)
= 3(x - 1)2
Note that (x – 1)2 > 0 for all x ∈ R, x ≠ 1.
∴ 3(x - 1)2 > 0 for all x ∈ R, x ≠ 1
∴ f(x) is increasing for all x ∈ R, x ≠ 1.
APPEARS IN
RELATED QUESTIONS
Find the value of c in Rolle's theorem for the function `f(x) = x^3 - 3x " in " (-sqrt3, 0)`
Prove that the function f(x) = loga x is increasing on (0, ∞) if a > 1 and decreasing on (0, ∞), if 0 < a < 1 ?
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) = x2 + 2x − 5 ?
Show that f(x) = cos x is a decreasing function on (0, π), increasing in (−π, 0) and neither increasing nor decreasing in (−π, π) ?
Show that f(x) = x9 + 4x7 + 11 is an increasing function for all x ∈ R ?
Find the intervals in which f(x) = log (1 + x) −\[\frac{x}{1 + x}\] is increasing or decreasing ?
Prove that the function f given by f(x) = x3 − 3x2 + 4x is strictly increasing on R ?
What are the values of 'a' for which f(x) = ax is decreasing on R ?
Write the set of values of 'a' for which f(x) = loga x is increasing in its domain ?
Write the set of values of k for which f(x) = kx − sin x is increasing on R ?
Write the set of values of a for which f(x) = cos x + a2 x + b is strictly increasing on R ?
Function f(x) = x3 − 27x + 5 is monotonically increasing when ______.
In the interval (1, 2), function f(x) = 2 | x − 1 | + 3 | x − 2 | is
The function f(x) = −x/2 + sin x defined on [−π/3, π/3] is
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.
The edge of a cube is decreasing at the rate of`( 0.6"cm")/sec`. Find the rate at which its volume is decreasing, when the edge of the cube is 2 cm.
Show that y = `log (1 + x) – (2x)/(2 + x), x > - 1` is an increasing function on its domain.
Test whether the function f(x) = x3 + 6x2 + 12x − 5 is increasing or decreasing for all x ∈ R
Choose the correct alternative:
The function f(x) = x3 – 3x2 + 3x – 100, x ∈ R is
Find the values of x such that f(x) = 2x3 – 15x2 – 144x – 7 is decreasing function
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] ______
If f(x) = `x^(3/2) (3x - 10)`, x ≥ 0, then f(x) is increasing in ______.
The interval on which the function f(x) = 2x3 + 9x2 + 12x – 1 is decreasing is ______.
The function f (x) = 2 – 3 x is ____________.
The function f(x) = x2 – 2x is increasing in the interval ____________.
Let f(x) = `x/sqrt(a^2 + x^2) - (d - x)/sqrt(b^2 + (d - x)^2), x ∈ R` where a, b and d are non-zero real constants. Then ______.
Find the interval in which the function f(x) = x2e–x is strictly increasing or decreasing.
