Advertisements
Advertisements
प्रश्न
Without using the derivative show that the function f (x) = 7x − 3 is strictly increasing function on R ?
Advertisements
उत्तर
\[\text { Here }, \]
\[f\left( x \right) = 7x - 3\]
\[\text { Let } x_1 , x_2 \text { in R such that } x_1 < x_2 . \text { Then },\]
\[ x_1 < x_2 \]
\[ \Rightarrow 7 x_1 < 7 x_2 \left[ \because 7 >0 \right]\]
\[ \Rightarrow 7 x_1 - 3 < 7 x_2 - 3\]
\[ \Rightarrow f\left( x_1 \right) < f\left( x_2 \right)\]
\[\therefore x_1 < x_2 \Rightarrow f\left( x_1 \right) < f\left( x_2 \right), \forall x_1 , x_2 \in R\]
\[\text { So,}f\left( x \right)\text { is strictly increasing on R } .\]
APPEARS IN
संबंधित प्रश्न
Find the intervals in which f(x) = sin 3x – cos 3x, 0 < x < π, is strictly increasing or strictly decreasing.
Find the value of c in Rolle's theorem for the function `f(x) = x^3 - 3x " in " (-sqrt3, 0)`
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:
−2x3 − 9x2 − 12x + 1
Find the intervals in which the following functions are strictly increasing or decreasing:
6 − 9x − x2
Prove that the function f(x) = loge x is increasing on (0, ∞) ?
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \left\{ x(x - 2) \right\}^2\] ?
Show that f(x) = e1/x, x ≠ 0 is a decreasing function for all x ≠ 0 ?
Show that f(x) = x2 − x sin x is an increasing function on (0, π/2) ?
Show that f(x) = x + cos x − a is an increasing function on R for all values of a ?
Let f defined on [0, 1] be twice differentiable such that | f (x) | ≤ 1 for all x ∈ [0, 1]. If f(0) = f(1), then show that | f'(x) | < 1 for all x ∈ [ 0, 1] ?
Find 'a' for which f(x) = a (x + sin x) + a is increasing on R ?
Write the interval in which f(x) = sin x + cos x, x ∈ [0, π/2] is increasing ?
Let f(x) = x3 + ax2 + bx + 5 sin2x be an increasing function on the set R. Then, a and b satisfy.
Every invertible function is
The function \[f\left( x \right) = \frac{\lambda \sin x + 2 \cos x}{\sin x + \cos x}\] is increasing, if
Function f(x) = ax is increasing on R, if
Let ϕ(x) = f(x) + f(2a − x) and f"(x) > 0 for all x ∈ [0, a]. Then, ϕ (x)
If the function f(x) = x2 − kx + 5 is increasing on [2, 4], then
The function f(x) = −x/2 + sin x defined on [−π/3, π/3] is
Show that the function f given by f(x) = tan–1 (sin x + cos x) is decreasing for all \[x \in \left( \frac{\pi}{4}, \frac{\pi}{2} \right) .\]
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. \[\left[ \text{ Use } \pi = \frac{22}{7} \right]\]
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).
Find the values of x for which the function f(x) = x3 – 12x2 – 144x + 13 (a) increasing (b) decreasing
Solve the following : Find the intervals on which the function y = xx, (x > 0) is increasing and decreasing.
Find the value of x, such that f(x) is decreasing function.
f(x) = 2x3 - 15x2 - 144x - 7
State whether the following statement is True or False:
If the function f(x) = x2 + 2x – 5 is an increasing function, then x < – 1
A circular pIate is contracting at the uniform rate of 5cm/sec. The rate at which the perimeter is decreasing when the radius of the circle is 10 cm Jong is
Let f(x) = x3 + 9x2 + 33x + 13, then f(x) is ______.
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 : ______.
Which of the following functions is decreasing on `(0, pi/2)`?
The function f(x) = tanx – x ______.
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(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 interval in which the function f(x) = `(4x^2 + 1)/x` is decreasing is ______.
Let f(x) = x3 – 6x2 + 9x + 18, then f(x) is strictly increasing in ______.
The intevral in which the function f(x) = 5 + 36x – 3x2 increases will be ______.
