Advertisements
Advertisements
प्रश्न
The function \[f\left( x \right) = \log_e \left( x^3 + \sqrt{x^6 + 1} \right)\] is of the following types:
विकल्प
even and increasing
odd and increasing
even and decreasing
odd and decreasing
Advertisements
उत्तर
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
संबंधित प्रश्न
The function f (x) = x3 – 3x2 + 3x – 100, x∈ R is _______.
(A) increasing
(B) decreasing
(C) increasing and decreasing
(D) neither increasing nor decreasing
Show that the function given by f(x) = 3x + 17 is strictly increasing on R.
Find the intervals in which the following functions are strictly increasing or decreasing:
(x + 1)3 (x − 3)3
Find the intervals in which the function f given by `f(x) = x^3 + 1/x^3 x != 0`, is (i) increasing (ii) decreasing.
Prove that f(x) = ax + b, where a, b are constants and a < 0 is a decreasing function on R ?
Show that f(x) = \[\frac{1}{1 + x^2}\] decreases in the interval [0, ∞) and increases in the interval (−∞, 0] ?
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) = 10 − 6x − 2x2 ?
Find the interval in which the following function are increasing or decreasing f(x) = x2 + 2x − 5 ?
Find the interval in which the following function are increasing or decreasing f(x) = 8 + 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 \[f\left( x \right) = \log\left( 2 + x \right) - \frac{2x}{2 + x}, x \in R\] ?
Show that f(x) = tan x is an increasing function on (−π/2, π/2) ?
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 interval in which f(x) = sin x + cos x, x ∈ [0, π/2] is increasing ?
The interval of increase of the function f(x) = x − ex + tan (2π/7) is
The function f(x) = cot−1 x + x increases in the interval
Let \[f\left( x \right) = \tan^{- 1} \left( g\left( x \right) \right),\],where g (x) is monotonically increasing for 0 < x < \[\frac{\pi}{2} .\] Then, f(x) is
In the interval (1, 2), function f(x) = 2 | x − 1 | + 3 | x − 2 | is
In the interval (1, 2), function f(x) = 2 | x − 1 | + 3 | x − 2 | 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.
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) = `x + (25)/x`
Find the values of x for which f(x) = `x/(x^2 + 1)` is (a) strictly increasing (b) decreasing.
Find the value of x, such that f(x) is decreasing function.
f(x) = 2x3 – 15x2 – 84x – 7
Show that f(x) = x – cos x is increasing for all x.
Find the values of x for which the function f(x) = 2x3 – 6x2 + 6x + 24 is strictly increasing
Choose the correct alternative:
The function f(x) = x3 – 3x2 + 3x – 100, x ∈ R is
If the function f(x) = `7/x - 3`, x ∈ R, x ≠ 0 is a decreasing function, then x ∈ ______
State whether the following statement is True or False:
The function f(x) = `3/x` + 10, x ≠ 0 is decreasing
A man of height 1.9 m walks directly away from a lamp of height 4.75m on a level road at 6m/s. The rate at which the length of his shadow is increasing is
If f(x) = [x], where [x] is the greatest integer not greater than x, then f'(1') = ______.
If f(x) = `x^(3/2) (3x - 10)`, x ≥ 0, then f(x) is increasing in ______.
In case of decreasing functions, slope of tangent and hence derivative is ____________.
Which of the following graph represent the strictly increasing function.
The function f(x) = `(4x^3 - 3x^2)/6 - 2sinx + (2x - 1)cosx` ______.
Let f: [0, 2]→R be a twice differentiable function such that f"(x) > 0, for all x ∈( 0, 2). If `phi` (x) = f(x) + f(2 – x), then `phi` 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 ______.
The function f(x) = tan–1(sin x + cos x) is an increasing function in ______.
The function f(x) = xex(1 − x), x ∈ R, is ______.
