Advertisements
Advertisements
Question
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] ?
Solution
\[\text { A function f(x) is said to be increasing on } \left[ a, b \right] \text { if it is increasing at x = a and x = b } . \]
\[\text { Here, } \]
\[f\left( x \right) = x^2 - 6x + 3\]
\[f'\left( x \right) = 2x - 6\]
\[ \Rightarrow f'\left( x \right) = 2\left( x - 3 \right)\]
\[\text { Now, } f'\left( 4 \right) = 2\left( 4 - 3 \right)\]
\[ = 2\]
\[ \therefore f'\left( 4 \right) > 0 \]
\[\text { So, f(x) is increasing on x} = 4 \]
\[\text { &, }f'\left( 6 \right) = 2\left( 6 - 3 \right)\]
\[ = 6\]
\[ \therefore f'\left( 6 \right) > 0 \]
\[\text { So, f (x) is increasing on x } = 6 \]
\[\text { Hence,}f\left( x \right)\text { is increasing on } [4, 6].\]
APPEARS IN
RELATED QUESTIONS
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 f(x) = sin 3x – cos 3x, 0 < x < π, is strictly increasing or strictly decreasing.
Find the intervals in which the function f given by f(x) = 2x2 − 3x is
- strictly increasing
- strictly decreasing
Show that y = `log(1+x) - (2x)/(2+x), x> - 1`, is an increasing function of x throughout its domain.
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) = x2 + 2x − 5 ?
Find the interval in which the following function are increasing or decreasing f(x) = 8 + 36x + 3x2 − 2x3 ?
Find the interval in which the following function are increasing or decreasing f(x) = 2x3 + 9x2 + 12x + 20 ?
Find the interval in which the following function are increasing or decreasing f(x) = x3 − 12x2 + 36x + 17 ?
Find the interval in which the following function are increasing or decreasing f(x) = \[5 x^\frac{3}{2} - 3 x^\frac{5}{2}\] x > 0 ?
Show that f(x) = log sin x is increasing on (0, π/2) and decreasing on (π/2, π) ?
Show that f(x) = (x − 1) ex + 1 is an increasing function for all x > 0 ?
Prove that the function f(x) = x3 − 6x2 + 12x − 18 is increasing on R ?
Show that f(x) = tan−1 x − x is a decreasing function on R ?
Prove that the function f given by f(x) = x − [x] is increasing in (0, 1) ?
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) ?
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] ?
Find the interval in which f(x) is increasing or decreasing f(x) = x|x|, x \[\in\] R ?
State whether f(x) = tan x − x is increasing or decreasing its domain ?
The function f(x) = x9 + 3x7 + 64 is increasing on
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
Find the intervals in which function f given by f(x) = 4x3 - 6x2 - 72x + 30 is (a) strictly increasing, (b) strictly decresing .
Find the values of x for which the following func- tions are strictly increasing : f(x) = x3 – 6x2 – 36x + 7
show that f(x) = `3x + (1)/(3x)` is increasing in `(1/3, 1)` and decreasing in `(1/9, 1/3)`.
Find the value of x, such that f(x) is decreasing function.
f(x) = 2x3 – 15x2 – 84x – 7
Let f(x) = x3 − 6x2 + 9𝑥 + 18, then f(x) is strictly decreasing in ______
Show that f(x) = x – cos x is increasing for all x.
Find the values of x for which the function f(x) = x3 – 6x2 – 36x + 7 is strictly increasing
If the function f(x) = `7/x - 3`, x ∈ R, x ≠ 0 is a decreasing function, then x ∈ ______
f(x) = `{{:(0"," x = 0 ), (x - 3"," x > 0):}` The function f(x) is ______
The sides of a square are increasing at the rate of 0.2 cm/sec. When the side is 25cm long, its area is increasing at the rate of ______
Show that f(x) = tan–1(sinx + cosx) is an increasing function in `(0, pi/4)`
The function f(x) = `(2x^2 - 1)/x^4`, x > 0, decreases in the interval ______.
The function f(x) = tan-1 x is ____________.
The function f(x) = x3 + 6x2 + (9 + 2k)x + 1 is strictly increasing for all x, if ____________.
The function `"f"("x") = "log" (1 + "x") - (2"x")/(2 + "x")` is increasing on ____________.
The function f(x) = tan–1(sin x + cos x) is an increasing function in ______.
