Find the intervals in which the following functions are increasin… - Vidyadip

Question

Find the intervals in which the following functions are increasing or decreasing.
f(x) = 2x³ - 15x² + 36x + 1

✓

Answer

f(x) = 2x³ - 15x² + 36x + 1
f'(x) = 6x² - 30x + 36
= 6(x² - 5x + 6)
= 6(x - 2)(x - 3)
For f(x) to be increasing, we must have
f'(x) > 0
⇒ 6(x - 2)(x - 3) > 0
⇒ (x - 2)(x - 3) > 0
[Since, 6 > 0, 6 (x - 2)(x - 3) > 0 ⇒ (x - 2)(x - 3) > 0]
⇒ x < 2 or x > 3
$\Rightarrow\text{x}\in(-\infty,2)\cup(3,\infty)$
So, f(x) is increasing on $\text{x}\in(-\infty,2)\cup(3,\infty).$
For f(x) to be decreasing, we must have,
f'(x) < 0
⇒ 6(x - 2)(x - 3) < 0
⇒ (x - 2)(x - 3) < 0
[Since, 6 > 0, 6 (x - 2)(x - 3) < 0 ⇒ (x - 2)(x - 3) < 0]
⇒ 2 < x < 3
$\Rightarrow\text{x}\in(2,3)$
So, f(x) is decreasing on $\text{x}\in(2,3).$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "4 Marks" from Increasing and Decreasing Functions→Increasing and Decreasing Functions — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Find:
$\int \frac{\cos \theta}{(4 + \sin^{2} \theta) (5 - 4 \cos^{2} \theta)} \text{d} \theta$

Solve the following differential equations:

$\cos\text{y}\frac{\text{dy}}{\text{dx}}=\text{e}^{\text{x}},\text{y}(0)=\frac{\pi}{2}$

The contents of three urns are as follows:

Urn 1 : 7 white, 3 black balls,

Urn 2 : 4 white, 6 black balls,

Urn 3 : 2 white, 8 black balls.

One of these urns is chosen at random with probabilities 0.20, 0.60 and 0.20 respectively. From the chosen urn two balls are drawn at random without replacement. If both these balls are white, what is the probability that these came from urn 3?

Find the area of the ragion in the first quadrant bounded by the parabola y = 4x² and the lines x = 0, y = 1 and y = 4.

Verify Lagrange's mean value theorem for the following function on the indicated intervals. find a point 'c' in the indicated interval as stated by the Lagrange's mean value theorem.
f(x) = x(x - 1) on [1, 2]

A bag contains 4 balls. Two balls are random (without replacement) and are found to be white. What is the probability that all balls in the bag are white?

Two cards are selected at random from a box which contains five cards numbered 1, 1, 2, 2, and 3. Let X denote the sum and Y the maximum of the two numbers drawn. Find the probability distribution, mean and variance of X and Y.

ABCD is aparallelogram. the position vectora of the points A, B and C are respectively, $4\hat{\text{i}}+5\hat{\text{j}}-10\hat{\text{k}},2\hat{\text{i}}-3\hat{\text{j}}+4\hat{\text{k}}$ and $-\hat{\text{i}}+2\hat{\text{j}}+\hat{\text{k}}.$ Find the vector equation of the line BD. Also, reduce it to cartesian form.

In order to supplement daily diet, a person wishes to take X and Y tablets. The contents (in milligrams per tablet) of iron, calcium and vitamins in X and Y are given as below:

Tablets	Iron	Calcium	Vitamin
X	6	3	2
Y	2	3	4

The person needs to supplement at least 18 milligrams of iron, 21 milligrams of calcium and 16 milligrams of vitamins. The price of each tablet of X and Y is ₹ 2 and ₹1 respectively. How many tablets of each type should the person take in order to satisfy the above requirement at the minimum cost? Make an LPP and solve graphically.

Coloured balls are distributed in four boxes as shown in the following table:

Box	Colour
Box	Black	White	Red	Blue
I II III IV	3 2 1 4	4 2 2 3	5 2 3 1	6 2 1 5

A box is selected at random and then a ball is randomly drawn from the selected box. The colour of the ball is black, what is the probability that ball drawn is from the box III.