Advertisements
Advertisements
Questions
Show that f(x) = e2x is increasing on R.
Show that the function given by f (x) = e 2x is increasing on R.
Advertisements
Solution 1
\[f\left( x \right) = e^{2x} \]
\[f'\left( x \right) = 2 e^{2x} \]
\[\text { Now,} \]
\[x \in R\]
Since the value of `e^{2x}` text is always positive for any real value of x, ` e^{2x}` > 0 .
\[ \Rightarrow 2 e^{2x} > 0\]
\[ \Rightarrow f'\left( x \right) > 0\]
\[\text { So,f(x)is increasing on R} .\]
Solution 2
We have f(x) = e2x
f'(x) = 2e2x > 0, x `in` R
f is strictly increasing on R
APPEARS IN
RELATED QUESTIONS
Find the intervals in which the function f(x) = 3x4 − 4x3 − 12x2 + 5 is
(a) strictly increasing
(b) strictly decreasing
Prove that f(x) = ax + b, where a, b are constants and a < 0 is a decreasing function on R ?
Without using the derivative show that the function f (x) = 7x − 3 is strictly increasing function on R ?
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 \[f\left( x \right) = \frac{3}{10} x^4 - \frac{4}{5} x^3 - 3 x^2 + \frac{36}{5}x + 11\] ?
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \frac{x^4}{4} + \frac{2}{3} x^3 - \frac{5}{2} x^2 - 6x + 7\] ?
Show that f(x) = tan−1 (sin x + cos x) is a decreasing function on the interval (π/4, π/2) ?
Show that f(x) = sin x − cos x is an increasing function on (−π/4, π/4)?
Show that the function f given by f(x) = 10x is increasing for all x ?
Find the interval in which f(x) is increasing or decreasing f(x) = sinx(1 + cosx), 0 < x < \[\frac{\pi}{2}\] ?
Write the set of values of 'a' for which f(x) = loga x is decreasing in its domain ?
Write the set of values of a for which f(x) = cos x + a2 x + b is strictly increasing on R ?
The function f(x) = xx decreases on the interval
The radius r of a right circular cylinder is increasing uniformly at the rate of 0·3 cm/s and its height h is decreasing at the rate of 0·4 cm/s. When r = 3·5 cm and h = 7 cm, find the rate of change of the curved surface area of the cylinder. \[\left[ \text{ Use } \pi = \frac{22}{7} \right]\]
Find `dy/dx,if e^x+e^y=e^(x-y)`
Prove that the function `f(x) = x^3- 6x^2 + 12x+5` is increasing on R.
Find the values of x for which the following functions are strictly increasing:
f(x) = 3 + 3x – 3x2 + x3
Show that f(x) = x – cos x is increasing for all x.
Solve the following:
Find the intervals on which the function f(x) = `x/logx` is increasing and decreasing.
Find the values of x for which the function f(x) = 2x3 – 6x2 + 6x + 24 is strictly increasing
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 ______.
State whether the following statement is True or False:
The function f(x) = `3/x` + 10, x ≠ 0 is decreasing
State whether the following statement is True or False:
If the function f(x) = x2 + 2x – 5 is an increasing function, then x < – 1
Find the values of x such that f(x) = 2x3 – 15x2 + 36x + 1 is increasing function
Show that for a ≥ 1, f(x) = `sqrt(3)` sinx – cosx – 2ax + b ∈ is decreasing in R
y = x(x – 3)2 decreases for the values of x given by : ______.
The function f(x) = tan-1 x is ____________.
Let `"f (x) = x – cos x, x" in "R"`, then f is ____________.
In `(0, pi/2),` the function f (x) = `"x"/"sin x"` is ____________.
2x3 - 6x + 5 is an increasing function, if ____________.
If f(x) = x3 + 4x2 + λx + 1(λ ∈ R) is a monotonically decreasing function of x in the largest possible interval `(–2, (–2)/3)` then ______.
If f(x) = x + cosx – a then ______.
y = log x satisfies for x > 1, the inequality ______.
Find the interval/s in which the function f : R `rightarrow` R defined by f(x) = xex, is increasing.
Let \[x_0\] be in the domain of a real-valued function \[f\]. When is \[f\] decreasing at \[x_0\]?
Which form shows that \[f'(x)=3x^2-6x+4\] is positive for every \[x\in\mathbf{R}\]?
