Advertisements
Advertisements
प्रश्न
Find the values of x for which the function f(x) = x3 – 6x2 – 36x + 7 is strictly increasing
Advertisements
उत्तर
f(x) = x3 – 6x2 – 36x + 7
∴ f′(x) = 3x2 – 12x – 36
= 3(x2 – 4x – 12)
= 3(x – 6)(x + 2)
f(x) is strictly increasing, if f′(x) > 0
∴ 3(x – 6)(x + 2) > 0
∴ (x – 6)(x + 2) > 0
ab > 0 ⇔ a > 0 and b > 0 or a < 0 and b < 0
Either x – 6 > 0 and x + 2 > 0
or
x – 6 < 0 and x + 2 < 0
Case I: x – 6 > 0 and x + 2 > 0
∴ x > 6 and x > – 2
∴ x > 6
Case II: x – 6 < 0 and x + 2 < 0
∴ x < 6 and x < – 2
∴ x < – 2
Thus, f(x) is strictly increasing for x ∈ (−∞ −2) ∪ (6, ∞).
APPEARS IN
संबंधित प्रश्न
The amount of pollution content added in air in a city due to x-diesel vehicles is given by P(x) = 0.005x3 + 0.02x2 + 30x. Find the marginal increase in pollution content when 3 diesel vehicles are added and write which value is indicated in the above question.
Find the intervals in which the function f given by f(x) = 2x3 − 3x2 − 36x + 7 is
- Strictly increasing
- Strictly decreasing
Prove that y = `(4sin theta)/(2 + cos theta) - theta` is an increasing function of θ in `[0, pi/2]`
Let I be any interval disjoint from (−1, 1). Prove that the function f given by `f(x) = x + 1/x` is strictly increasing on I.
Prove that f(x) = ax + b, where a, b are constants and a > 0 is an increasing function on R ?
Prove that f(x) = ax + b, where a, b are constants and a < 0 is a decreasing function on R ?
Find the interval in which the following function are increasing or decreasing f(x) = 2x3 − 15x2 + 36x + 1 ?
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) = sin x is increasing on (0, π/2) and decreasing on (π/2, π) and neither increasing nor decreasing in (0, π) ?
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) ?
Determine whether f(x) = −x/2 + sin x is increasing or decreasing on (−π/3, π/3) ?
Show that the function f given by f(x) = 10x is increasing for all x ?
Find the value(s) of a for which f(x) = x3 − ax is an increasing function on R ?
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] ?
What are the values of 'a' for which f(x) = ax is decreasing on R ?
Find the values of 'a' for which the function f(x) = sin x − ax + 4 is increasing function on R ?
The function f(x) = 2 log (x − 2) − x2 + 4x + 1 increases on the interval
f(x) = 2x − tan−1 x − log \[\left\{ x + \sqrt{x^2 + 1} \right\}\] is monotonically increasing when
If the function f(x) = x2 − kx + 5 is increasing on [2, 4], then
The function f(x) = x9 + 3x7 + 64 is increasing on
Find the intervals in which function f given by f(x) = 4x3 - 6x2 - 72x + 30 is (a) strictly increasing, (b) strictly decresing .
For manufacturing x units, labour cost is 150 – 54x and processing cost is x2. Price of each unit is p = 10800 – 4x2. Find the value of x for which Total cost is decreasing.
Test whether the following functions are increasing or decreasing : f(x) = 2 – 3x + 3x2 – x3, x ∈ R.
Test whether the following functions are increasing or decreasing: f(x) = `x-(1)/x`, x ∈ R, x ≠ 0.
Show that function f(x) =`("x - 2")/("x + 1")`, x ≠ -1 is increasing.
Let f(x) = x3 − 6x2 + 9𝑥 + 18, then f(x) is strictly decreasing in ______
Find the values of x for which the function f(x) = 2x3 – 6x2 + 6x + 24 is strictly increasing
Find the values of x, for which the function f(x) = x3 + 12x2 + 36𝑥 + 6 is monotonically decreasing
In which interval is the given function, f(x) = 2x3 - 21x2 + 72x + 19 monotonically decreasing?
If f(x) = x3 – 15x2 + 84x – 17, then ______.
Let the f : R → R be defined by f (x) = 2x + cosx, then f : ______.
Which of the following functions is decreasing on `(0, pi/2)`?
The function f(x) = tanx – x ______.
The function f(x) = `(2x^2 - 1)/x^4`, x > 0, decreases in the interval ______.
In case of decreasing functions, slope of tangent and hence derivative is ____________.
The function f (x) = 2 – 3 x is ____________.
Let `"f (x) = x – cos x, x" in "R"`, then f is ____________.
`"f"("x") = (("e"^(2"x") - 1)/("e"^(2"x") + 1))` is ____________.
The length of the longest interval, in which the function `3 "sin x" - 4 "sin"^3"x"` is increasing, is ____________.
The function f(x) = `|x - 1|/x^2` is monotonically decreasing on ______.
Find the interval/s in which the function f : R `rightarrow` R defined by f(x) = xex, is increasing.
Find the values of x for which the function f(x) = `x/(x^2 + 1)` is strictly decreasing.
The function f(x) = xex(1 − x), x ∈ R, is ______.
Let \[x_0\] be in the domain of a real-valued function \[f\]. When is \[f\] increasing at \[x_0\]?
For \[f(x)=4x^3-6x^2-72x+30\], which factorization of \[f'(x)\] is correct?
