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
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
Price P for demand D is given as P = 183 +120D - 3D2 Find D for which the price is increasing
Find the intervals in which the function f(x) = 3x4 − 4x3 − 12x2 + 5 is
(a) strictly increasing
(b) strictly decreasing
The function f (x) = x3 – 3x2 + 3x – 100, x∈ R is _______.
(A) increasing
(B) decreasing
(C) increasing and decreasing
(D) neither increasing nor decreasing
Find the intervals in which the following functions are strictly increasing or decreasing:
x2 + 2x − 5
Find the intervals in which the following functions are strictly increasing or decreasing:
10 − 6x − 2x2
Find the intervals in which the following functions are strictly increasing or decreasing:
6 − 9x − x2
Prove that the function f given by f(x) = x2 − x + 1 is neither strictly increasing nor strictly decreasing on (−1, 1).
On which of the following intervals is the function f given byf(x) = x100 + sin x –1 strictly decreasing?
Prove that the function given by f (x) = x3 – 3x2 + 3x – 100 is increasing in R.
Prove that the function f(x) = loga x is increasing on (0, ∞) if a > 1 and decreasing on (0, ∞), if 0 < a < 1 ?
Find the interval in which the following function are increasing or decreasing f(x) = 2x3 − 9x2 + 12x − 5 ?
Find the interval in which the following function are increasing or decreasing f(x) = 2x3 − 24x + 107 ?
Find the interval in which the following function are increasing or decreasing f(x) = x3 − 12x2 + 36x + 17 ?
Show that f(x) = x3 − 15x2 + 75x − 50 is an increasing function for all x ∈ R ?
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 the function x2 − x + 1 is neither increasing nor decreasing on (0, 1) ?
Find the values of b for which the function f(x) = sin x − bx + c is a decreasing function on R ?
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 ?
Let f(x) = x3 + ax2 + bx + 5 sin2x be an increasing function on the set R. Then, a and b satisfy.
The function \[f\left( x \right) = \frac{\lambda \sin x + 2 \cos x}{\sin x + \cos x}\] is increasing, if
Find the intervals in which function f given by f(x) = 4x3 - 6x2 - 72x + 30 is (a) strictly increasing, (b) strictly decresing .
Find the values of x for which the following functions are strictly increasing : f(x) = 2x3 – 3x2 – 12x + 6
Find the values of x for which the following functions are strictly decreasing : f(x) = x3 – 9x2 + 24x + 12
Show that y = `log (1 + x) – (2x)/(2 + x), x > - 1` is an increasing function on its domain.
Show that function f(x) =`3/"x" + 10`, x ≠ 0 is decreasing.
Find the values of x for which the function f(x) = x3 – 6x2 – 36x + 7 is strictly increasing
The slope of tangent at any point (a, b) is also called as ______.
If f(x) = x3 – 15x2 + 84x – 17, then ______.
y = x(x – 3)2 decreases for the values of x given by : ______.
The function f(x) = x2 – 2x is increasing in the interval ____________.
The function which is neither decreasing nor increasing in `(pi/2,(3pi)/2)` is ____________.
The function `"f"("x") = "log" (1 + "x") - (2"x")/(2 + "x")` is increasing on ____________.
The function f: N → N, where
f(n) = `{{:(1/2(n + 1), "If n is sold"),(1/2n, "if n is even"):}` 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 ______.
y = log x satisfies for x > 1, the inequality ______.
The interval in which the function f(x) = `(4x^2 + 1)/x` is decreasing is ______.
In which one of the following intervals is the function f(x) = x3 – 12x increasing?
