Advertisements
Advertisements
Question
Find the values of x for which the function f(x) = x3 – 6x2 – 36x + 7 is strictly increasing
Advertisements
Solution
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
RELATED QUESTIONS
Find the intervals in which the function f(x) = 3x4 − 4x3 − 12x2 + 5 is
(a) strictly increasing
(b) strictly decreasing
Find the value(s) of x for which y = [x(x − 2)]2 is an increasing function.
Find the values of x for `y = [x(x - 2)]^2` is an increasing function.
Prove that y = `(4sin theta)/(2 + cos theta) - theta` is an increasing function of θ in `[0, pi/2]`
Prove that the function given by f (x) = x3 – 3x2 + 3x – 100 is increasing in R.
Prove that f(x) = ax + b, where a, b are constants and a > 0 is an increasing function on R ?
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) = x3 − 6x2 − 36x + 2 ?
Find the interval in which the following function are increasing or decreasing f(x) = x4 − 4x3 + 4x2 + 15 ?
Determine the values of x for which the function f(x) = x2 − 6x + 9 is increasing or decreasing. Also, find the coordinates of the point on the curve y = x2 − 6x + 9 where the normal is parallel to the line y = x + 5 ?
Show that f(x) = e1/x, x ≠ 0 is a decreasing function for all x ≠ 0 ?
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) ?
State when a function f(x) is said to be increasing on an interval [a, b]. Test whether the function f(x) = x2 − 6x + 3 is increasing on the interval [4, 6] ?
Prove that the function f given by f(x) = log cos x is strictly increasing on (−π/2, 0) and strictly decreasing on (0, π/2) ?
Find 'a' for which f(x) = a (x + sin x) + a is increasing on R ?
If g (x) is a decreasing function on R and f(x) = tan−1 [g (x)]. State whether f(x) is increasing or decreasing on R ?
Let f(x) = x3 + ax2 + bx + 5 sin2x be an increasing function on the set R. Then, a and b satisfy.
The function \[f\left( x \right) = \log_e \left( x^3 + \sqrt{x^6 + 1} \right)\] is of the following types:
In the interval (1, 2), function f(x) = 2 | x − 1 | + 3 | x − 2 | is
The function f(x) = −x/2 + sin x defined on [−π/3, π/3] is
Find the intervals in which the function \[f(x) = \frac{3}{2} x^4 - 4 x^3 - 45 x^2 + 51\] is
(a) strictly increasing
(b) strictly decreasing
Test whether the following functions are increasing or decreasing : f(x) = 2 – 3x + 3x2 – x3, x ∈ R.
Show that the function f(x) = x3 + 10x + 7 for x ∈ R is strictly increasing
By completing the following activity, find the values of x such that f(x) = 2x3 – 15x2 – 84x – 7 is decreasing function.
Solution: f(x) = 2x3 – 15x2 – 84x – 7
∴ f'(x) = `square`
∴ f'(x) = 6`(square) (square)`
Since f(x) is decreasing function.
∴ f'(x) < 0
Case 1: `(square)` > 0 and (x + 2) < 0
∴ x ∈ `square`
Case 2: `(square)` < 0 and (x + 2) > 0
∴ x ∈ `square`
∴ f(x) is decreasing function if and only if x ∈ `square`
A man of height 1.9 m walks directly away from a lamp of height 4.75m on a level road at 6m/s. The rate at which the length of his shadow is increasing is
f(x) = `{{:(0"," x = 0 ), (x - 3"," x > 0):}` The function f(x) is ______
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] ______
If f(x) = x3 – 15x2 + 84x – 17, then ______.
Show that f(x) = 2x + cot–1x + `log(sqrt(1 + x^2) - x)` is increasing in R
Let the f : R → R be defined by f (x) = 2x + cosx, then f : ______.
The function f(x) = 4 sin3x – 6 sin2x + 12 sinx + 100 is strictly ______.
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 ____________.
The function f(x) = mx + c where m, c are constants, is a strict decreasing function for all `"x" in "R"` , if ____________.
2x3 - 6x + 5 is an increasing function, if ____________.
The function `"f"("x") = "x"/"logx"` increases on the interval
Let h(x) = f(x) - [f(x)]2 + [f(x)]3 for every real number x. Then ____________.
Which of the following graph represent the strictly increasing function.
Show that function f(x) = tan x is increasing in `(0, π/2)`.
If f(x) = x3 + 4x2 + λx + 1(λ ∈ R) is a monotonically decreasing function of x in the largest possible interval `(–2, (–2)/3)` then ______.
y = log x satisfies for x > 1, the inequality ______.
Function f(x) = x100 + sinx – 1 is increasing for all x ∈ ______.
If f(x) = `x/(x^2 + 1)` is increasing function then the value of x lies in ______.
A function f is said to be increasing at a point c if ______.
The function f(x) = x3 + 3x is increasing in interval ______.
Find the interval/s in which the function f : R `rightarrow` R defined by f(x) = xex, is increasing.
The function f(x) = sin4x + cos4x is an increasing function if ______.
The intevral in which the function f(x) = 5 + 36x – 3x2 increases will be ______.
