Advertisements
Advertisements
प्रश्न
Prove that the function f(x) = x3 − 6x2 + 12x − 18 is increasing on R ?
Advertisements
उत्तर
\[f\left( x \right) = x^3 - 6 x^2 + 12x - 18\]
\[f'\left( x \right) = 3 x^2 - 12x + 12\]
\[ = 3\left( x^2 - 4x + 4 \right)\]
\[ = 3 \left( x - 2 \right)^2 \geq 0, \forall x \text { in R } \left[ \because 3 > 0 \text { &} \left( x - 2 \right)^2 \geq 0 \right]\]
\[\text { So, f(x) is increasing on R } .\]
APPEARS IN
संबंधित प्रश्न
Find the value(s) of x for which y = [x(x − 2)]2 is an increasing function.
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
On which of the following intervals is the function f given byf(x) = x100 + sin x –1 strictly decreasing?
Prove that the function f given by f(x) = log sin x is strictly increasing on `(0, pi/2)` and strictly decreasing on `(pi/2, pi)`
Show that the function f(x) = 4x3 - 18x2 + 27x - 7 is always increasing on R.
Find the intervals in which the function `f(x) = x^4/4 - x^3 - 5x^2 + 24x + 12` is (a) strictly increasing, (b) strictly decreasing
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(x) = 2x3 − 24x + 107 ?
Show that f(x) = log sin x is increasing on (0, π/2) and decreasing on (π/2, π) ?
Show that f(x) = x3 − 15x2 + 75x − 50 is an increasing function for all x ∈ R ?
Show that the function x2 − x + 1 is neither increasing nor decreasing on (0, 1) ?
Show that f(x) = x9 + 4x7 + 11 is an increasing function for all x ∈ R ?
Determine whether f(x) = −x/2 + sin x is increasing or decreasing on (−π/3, π/3) ?
Find the interval in which f(x) is increasing or decreasing f(x) = x|x|, x \[\in\] R ?
Find 'a' for which f(x) = a (x + sin x) + a 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 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. \[\left[ \text{ Use } \pi = \frac{22}{7} \right]\]
Prove that the function `f(x) = x^3- 6x^2 + 12x+5` is increasing on R.
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 increasing:
f(x) = 3 + 3x – 3x2 + x3
Find the values of x for which the function f(x) = x3 – 6x2 – 36x + 7 is strictly increasing
Show that the function f(x) = `(x - 2)/(x + 1)`, x ≠ – 1 is increasing
For which interval the given function f(x) = 2x3 – 9x2 + 12x + 7 is increasing?
If f(x) = `x^(3/2) (3x - 10)`, x ≥ 0, then f(x) is increasing in ______.
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) = 2 – 3 x is ____________.
Let f (x) = tan x – 4x, then in the interval `[- pi/3, pi/3], "f"("x")` is ____________.
The function which is neither decreasing nor increasing in `(pi/2,(3pi)/2)` is ____________.
The function f(x) = x3 + 6x2 + (9 + 2k)x + 1 is strictly increasing for all x, if ____________.
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`
Let 'a' be a real number such that the function f(x) = ax2 + 6x – 15, x ∈ R is increasing in `(-∞, 3/4)` and decreasing in `(3/4, ∞)`. Then the function g(x) = ax2 – 6x + 15, x∈R has a ______.
y = log x satisfies for x > 1, the inequality ______.
The interval in which the function f(x) = `(4x^2 + 1)/x` is decreasing is ______.
A function f is said to be increasing at a point c if ______.
The function f(x) = x3 + 3x is increasing in interval ______.
The intevral in which the function f(x) = 5 + 36x – 3x2 increases will be ______.
