Advertisements
Advertisements
प्रश्न
Find the value of x, such that f(x) is increasing function.
f(x) = 2x3 - 15x2 + 36x + 1
Advertisements
उत्तर
f(x) = 2x3 - 15x2 + 36x + 1
∴ f'(x) = 6x2 - 30x + 36
= 6(x2 - 5x + 6)
= 6(x - 3)(x - 2)
f(x) is an increasing function, if f'(x) > 0
∴ 6(x - 3)(x - 2) > 0
∴ (x - 3)(x - 2) > 0
ab > 0 ⇔ a > 0 and b > 0 or a < 0 or b < 0
∴ Either (x – 3) > 0 and (x – 2) > 0 or
(x – 3) < 0 and (x – 2) < 0
Case 1: x – 3 > 0 and x – 2 > 0
∴ x > 3 and x > 2
∴ x > 3
Case 2: x – 3 < 0 and x – 2 < 0
∴ x < 3 and x < 2
∴ x < 2
Thus, f(x) is an increasing function for x < 2 or x > 3, i.e., (- ∞, 2) ∪ (3, ∞)
APPEARS IN
संबंधित प्रश्न
Find the intervals in which the function f given by f(x) = 2x3 − 3x2 − 36x + 7 is
- Strictly increasing
- Strictly decreasing
Let f be a function defined on [a, b] such that f '(x) > 0, for all x ∈ (a, b). Then prove that f is an increasing function on (a, b).
Show that f(x) = \[\frac{1}{x}\] is a decreasing function on (0, ∞) ?
Find the interval in which the following function are increasing or decreasing f(x) = x3 − 6x2 + 9x + 15 ?
Find the intervals in which f(x) = (x + 2) e−x is increasing or decreasing ?
Find the value(s) of a for which f(x) = x3 − ax is an increasing function on R ?
Find the interval in which f(x) is increasing or decreasing f(x) = sinx + |sin x|, 0 < x \[\leq 2\pi\] ?
If the function f(x) = 2x2 − kx + 5 is increasing on [1, 2], then k lies in the interval
Let \[f\left( x \right) = \tan^{- 1} \left( g\left( x \right) \right),\],where g (x) is monotonically increasing for 0 < x < \[\frac{\pi}{2} .\] Then, f(x) is
If the function f(x) = kx3 − 9x2 + 9x + 3 is monotonically increasing in every interval, then
Prove that the function `f(x) = x^3- 6x^2 + 12x+5` is increasing on R.
Find the intervals in which function f given by f(x) = 4x3 - 6x2 - 72x + 30 is (a) strictly increasing, (b) strictly decresing .
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) = 2x3 – 3x2 – 12x + 6
Show that y = `log (1 + x) – (2x)/(2 + x), x > - 1` is an increasing function on its domain.
Find the value of x, such that f(x) is increasing function.
f(x) = 2x3 - 15x2 - 144x - 7
For manufacturing x units, labour cost is 150 – 54x and processing cost is x2. Price of each unit is p = 10800 – 4x2. Find the values of x for which Revenue is increasing.
Find the values of x such that f(x) = 2x3 – 15x2 + 36x + 1 is increasing function
Given P(x) = x4 + ax3 + bx2 + cx + d such that x = 0 is the only real root of P'(x) = 0. If P(-1) < P(1), then in the interval [-1, 1] ______
Show that f(x) = 2x + cot–1x + `log(sqrt(1 + x^2) - x)` is increasing in R
If f(x) = sin x – cos x, then interval in which function is decreasing in 0 ≤ x ≤ 2 π, is:
Find the interval in which the function `f` is given by `f(x) = 2x^2 - 3x` is strictly decreasing.
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 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 ______.
If f(x) = x5 – 20x3 + 240x, then f(x) satisfies ______.
Let f : R `rightarrow` R be a positive increasing function with `lim_(x rightarrow ∞) (f(3x))/(f(x))` = 1 then `lim_(x rightarrow ∞) (f(2x))/(f(x))` = ______.
A function \[f\] is monotonic in an interval when it is which of the following?
Let \[x_0\] be in the domain of a real-valued function \[f\]. When is \[f\] decreasing at \[x_0\]?
For \[f(x)=4x^3-6x^2-72x+30\], which factorization of \[f'(x)\] is correct?
