Advertisements
Advertisements
प्रश्न
Find the set of values of 'b' for which f(x) = b (x + cos x) + 4 is decreasing on R ?
उत्तर
\[f\left( x \right) = b\left( x + \cos x \right) + 4\]
\[f'\left( x \right) = b\left( 1 - \sin x \right)\]
\[\text { Given }:f(x) \text { is decreasing on R }.\]
\[ \Rightarrow f'\left( x \right) < 0\]
\[ \Rightarrow b\left( 1 - \sin x \right) < 0 . . . \left( 1 \right)\]
\[\text { We know },\]
\[\sin x \leq 1\]
\[ \Rightarrow 1 - \sin x \geq 0\]
\[ \Rightarrow b < 0 \left[ \text { Since } \left( 1 - \sin x \right) \geq 0, b\left( 1 - \sin x \right) < 0 \Rightarrow b < 0 \right]\]
\[ \Rightarrow b \in \left( - \infty , 0 \right)\]
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)`
Show that the function `f(x) = x^3 - 3x^2 + 6x - 100` is increasing on R
Test whether the function is increasing or decreasing.
f(x) = `"x" -1/"x"`, x ∈ R, x ≠ 0,
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:
x2 + 2x − 5
Find the intervals in which the following functions are strictly increasing or decreasing:
−2x3 − 9x2 − 12x + 1
Prove that y = `(4sin theta)/(2 + cos theta) - theta` is an increasing function of θ in `[0, pi/2]`
Find the intervals in which the function f given by `f(x) = (4sin x - 2x - x cos x)/(2 + cos x)` is (i) increasing (ii) decreasing.
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 the function f(x) = loga x is increasing on (0, ∞) if a > 1 and decreasing on (0, ∞), if 0 < a < 1 ?
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) = −2x3 − 9x2 − 12x + 1 ?
Show that f(x) = e1/x, x ≠ 0 is a decreasing function for all x ≠ 0 ?
Show that f(x) = tan x is an increasing function on (−π/2, π/2) ?
Show that f(x) = tan−1 (sin x + cos x) is a decreasing function on the interval (π/4, π/2) ?
Show that the function f(x) = cot \[-\] l(sinx + cosx) is decreasing on \[\left( 0, \frac{\pi}{4} \right)\] and increasing on \[\left( 0, \frac{\pi}{4} \right)\] ?
Show that f(x) = sin x − cos x is an increasing function on (−π/4, π/4) ?
Show that f(x) = tan−1 x − x is a decreasing function on R ?
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 ?
What are the values of 'a' for which f(x) = ax is increasing on R ?
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 the function f(x) = x3 – 12x2 – 144x + 13 (a) increasing (b) decreasing
Choose the correct alternative:
The function f(x) = x3 – 3x2 + 3x – 100, x ∈ R is
State whether the following statement is True or False:
The function f(x) = `3/x` + 10, x ≠ 0 is decreasing
The area of the square increases at the rate of 0.5 cm2/sec. The rate at which its perimeter is increasing when the side of the square is 10 cm long is ______.
f(x) = `{{:(0"," x = 0 ), (x - 3"," x > 0):}` The function f(x) is ______
For which interval the given function f(x) = 2x3 – 9x2 + 12x + 7 is increasing?
`"f"("x") = (("e"^(2"x") - 1)/("e"^(2"x") + 1))` is ____________.
Which of the following graph represent the strictly increasing function.
Let x0 be a point in the domain of definition of a real valued function `f` and there exists an open interval I = (x0 – h, ro + h) containing x0. Then which of the following statement is/ are true for the above statement.
Find the value of x for which the function f(x)= 2x3 – 9x2 + 12x + 2 is decreasing.
Given f(x) = 2x3 – 9x2 + 12x + 2
∴ f'(x) = `squarex^2 - square + square`
∴ f'(x) = `6(x - 1)(square)`
Now f'(x) < 0
∴ 6(x – 1)(x – 2) < 0
Since ab < 0 ⇔a < 0 and b < 0 or a > 0 and b < 0
Case 1: (x – 1) < 0 and (x – 2) < 0
∴ x < `square` and x > `square`
Which is contradiction
Case 2: x – 1 and x – 2 < 0
∴ x > `square` and x < `square`
1 < `square` < 2
f(x) is decreasing if and only if x ∈ `square`
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) = tan–1`phi`(x), where `phi`(x) is monotonically increasing for `0 < x < π/2`. Then f(x) is ______.
The intevral in which the function f(x) = 5 + 36x – 3x2 increases will be ______.
