Advertisements
Advertisements
प्रश्न
Prove that the function f(x) = x3 − 6x2 + 12x − 18 is increasing on R ?
उत्तर
\[f\left( x \right) = x^3 - 6 x^2 + 12x - 18\]
\[f'\left( x \right) = 3 x^2 - 12x + 12\]
\[ = 3\left( x^2 - 4x + 4 \right)\]
\[ = 3 \left( x - 2 \right)^2 \geq 0, \forall x \text { in R } \left[ \because 3 > 0 \text { &} \left( x - 2 \right)^2 \geq 0 \right]\]
\[\text { So, f(x) is increasing on R } .\]
APPEARS IN
संबंधित प्रश्न
Which of the following functions are strictly decreasing on `(0, pi/2)`?
- cos x
- cos 2x
- cos 3x
- tan x
Find the least value of a such that the function f given by f (x) = x2 + ax + 1 is strictly increasing on [1, 2].
The interval in which y = x2 e–x is increasing is ______.
Prove that the function f(x) = loga x is increasing on (0, ∞) if a > 1 and decreasing on (0, ∞), if 0 < a < 1 ?
Show that f(x) = \[\frac{1}{1 + x^2}\] is neither increasing nor decreasing on R ?
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) = 5 + 36x + 3x2 − 2x3 ?
Find the interval in which the following function are increasing or decreasing f(x) = 2x3 + 9x2 + 12x + 20 ?
Show that f(x) = log sin x is increasing on (0, π/2) and decreasing on (π/2, π) ?
Show that the function f given by f(x) = 10x is increasing for all x ?
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) ?
Show that f(x) = x2 − x sin x is an increasing function on (0, π/2) ?
What are the values of 'a' for which f(x) = ax is decreasing on R ?
Write the interval in which f(x) = sin x + cos x, x ∈ [0, π/2] is increasing ?
Function f(x) = x3 − 27x + 5 is monotonically increasing when
The function \[f\left( x \right) = \frac{\lambda \sin x + 2 \cos x}{\sin x + \cos x}\] is increasing, if
For manufacturing x units, labour cost is 150 – 54x and processing cost is x2. Price of each unit is p = 10800 – 4x2. Find the value of x for which Total cost is decreasing.
Test whether the following functions are increasing or decreasing : f(x) = x3 – 6x2 + 12x – 16, x ∈ R.
Test whether the following functions are increasing or decreasing : f(x) = 2 – 3x + 3x2 – x3, x ∈ R.
Find the values of x for which f(x) = `x/(x^2 + 1)` is (a) strictly increasing (b) decreasing.
Find the values of x for which f(x) = 2x3 – 15x2 – 144x – 7 is
(a) Strictly increasing
(b) strictly decreasing
The price P for the demand D is given as P = 183 + 120D − 3D2, then the value of D for which price is increasing, is ______.
The slope of tangent at any point (a, b) is also called as ______.
Find the values of x such that f(x) = 2x3 – 15x2 + 36x + 1 is increasing function
The function f(x) = 9 - x5 - x7 is decreasing for
f(x) = `{{:(0"," x = 0 ), (x - 3"," x > 0):}` The function f(x) is ______
For which interval the given function f(x) = 2x3 – 9x2 + 12x + 7 is increasing?
Let f be a real valued function defined on (0, 1) ∪ (2, 4) such that f '(x) = 0 for every x, then ____________.
In `(0, pi/2),` the function f (x) = `"x"/"sin x"` is ____________.
Which of the following graph represent the strictly increasing function.
The interval in which `y = x^2e^(-x)` is increasing with respect to `x` is
Let f(x) be a function such that; f'(x) = log1/3(log3(sinx + a)) (where a ∈ R). If f(x) is decreasing for all real values of x then the exhaustive solution set of a is ______.
Function f(x) = `log(1 + x) - (2x)/(2 + x)` is monotonically increasing when ______.
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))` = ______.
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 ______.
If f(x) = `x/(x^2 + 1)` is increasing function then the value of x lies in ______.
