Advertisements
Advertisements
Question
The function \[f\left( x \right) = \log_e \left( x^3 + \sqrt{x^6 + 1} \right)\] is of the following types:
Options
even and increasing
odd and increasing
even and decreasing
odd and decreasing
Solution
odd and increasing
\[f(x) = \log_e \left( x^3 + \sqrt{x^6 + 1} \right)\]
\[ \Rightarrow f( - x) = \log_e \left( - x^3 + \sqrt{x^6 + 1} \right)\]
\[ = \log_e \left\{ \frac{\left( - x^3 + \sqrt{x^6 + 1} \right)\left( x^3 + \sqrt{x^6 + 1} \right)}{x^3 + \sqrt{x^6 + 1}} \right\}\]
\[ = \log_e \left( \frac{x^6 + 1 - x^6}{x^3 + \sqrt{x^6 + 1}} \right)\]
\[ = \log_e \left( \frac{1}{x^3 + \sqrt{x^6 + 1}} \right)\]
\[ = - \log_e \left( x^3 + \sqrt{x^6 + 1} \right)\]
\[ = - f(x) \]
\[\text { Hence,} f( - x) = - f(x)\]
\[\text { Therefore, it is an odd function } .\]
\[f(x) = \log_e \left( x^3 + \sqrt{x^6 + 1} \right)\]
\[\frac{d}{dx}\left\{ f(x) \right\} = \left( \frac{1}{x^3 + \sqrt{x^6 + 1}} \right) \times \left( 3 x^2 + \frac{1}{2\sqrt{x^6 + 1}} \times 6 x^5 \right)\]
\[ = \left( \frac{1}{x^3 + \sqrt{x^6 + 1}} \right) \times \left( \frac{6 x^2 \sqrt{x^6 + 1} + 6 x^5}{2\sqrt{x^6 + 1}} \right)\]
\[ = \left( \frac{1}{x^3 + \sqrt{x^6 + 1}} \right) \times \left\{ \frac{6 x^2 \left( \sqrt{x^6 + 1} + x^3 \right)}{2\sqrt{x^6 + 1}} \right\}\]
\[ = \left( \frac{6 x^2}{2\sqrt{x^6 + 1}} \right) > 0\]
\[\text { Therefore, given function is an increasing function } .\]
APPEARS IN
RELATED QUESTIONS
The amount of pollution content added in air in a city due to x-diesel vehicles is given by P(x) = 0.005x3 + 0.02x2 + 30x. Find the marginal increase in pollution content when 3 diesel vehicles are added and write which value is indicated in the above question.
Find the intervals in which the function f given by `f(x) = x^3 + 1/x^3 x != 0`, is (i) increasing (ii) decreasing.
Find the intervals in which the function `f(x) = x^4/4 - x^3 - 5x^2 + 24x + 12` is (a) strictly increasing, (b) strictly decreasing
Water is dripping out from a conical funnel of semi-verticle angle `pi/4` at the uniform rate of `2 cm^2/sec`in the surface, through a tiny hole at the vertex of the bottom. When the slant height of the water level is 4 cm, find the rate of decrease of the slant height of the water.
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) = (x − 1) (x − 2)2 ?
Find the interval in which the following function are increasing or decreasing f(x) = 2x3 − 24x + 7 ?
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \frac{3}{10} x^4 - \frac{4}{5} x^3 - 3 x^2 + \frac{36}{5}x + 11\] ?
Show that f(x) = tan x is an increasing function on (−π/2, π/2) ?
Show that the function x2 − x + 1 is neither increasing nor decreasing on (0, 1) ?
State when a function f(x) is said to be increasing on an interval [a, b]. Test whether the function f(x) = x2 − 6x + 3 is increasing on the interval [4, 6] ?
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 ?
Find 'a' for which f(x) = a (x + sin x) + a is increasing on R ?
Find the set of values of 'b' for which f(x) = b (x + cos x) + 4 is decreasing on R ?
The function f(x) = 2 log (x − 2) − x2 + 4x + 1 increases on the interval
The price P for demand D is given as P = 183 + 120 D – 3D2.
Find D for which the price is increasing.
Using truth table show that ∼ (p → ∼ q) ≡ p ∧ q
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.
Find the values of x for which the following functions are strictly decreasing:
f(x) = 2x3 – 3x2 – 12x + 6
Show that f(x) = x – cos x is increasing for all x.
Show that y = `log (1 + x) – (2x)/(2 + x), x > - 1` is an increasing function on its domain.
Find the value of x, such that f(x) is increasing function.
f(x) = 2x3 - 15x2 + 36x + 1
Find the value of x, such that f(x) is decreasing function.
f(x) = 2x3 - 15x2 - 144x - 7
Find the value of x such that f(x) is decreasing function.
f(x) = x4 − 2x3 + 1
Show that function f(x) =`("x - 2")/("x + 1")`, x ≠ -1 is increasing.
Find the values of x for which the function f(x) = 2x3 – 6x2 + 6x + 24 is strictly increasing
State whether the following statement is True or False:
If the function f(x) = x2 + 2x – 5 is an increasing function, then x < – 1
The function f(x) = 9 - x5 - x7 is decreasing for
For every value of x, the function f(x) = `1/"a"^x`, a > 0 is ______.
If f(x) = x3 – 15x2 + 84x – 17, then ______.
The function f(x) = `(2x^2 - 1)/x^4`, x > 0, decreases in the interval ______.
In `(0, pi/2),` the function f (x) = `"x"/"sin x"` is ____________.
The function f(x) = tan-1 (sin x + cos x) is an increasing function in:
The interval in which `y = x^2e^(-x)` is increasing with respect to `x` is
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))` = ______.
The function f(x) = x3 + 3x is increasing in interval ______.
The intevral in which the function f(x) = 5 + 36x – 3x2 increases will be ______.
Find the interval in which the function f(x) = x2e–x is strictly increasing or decreasing.
