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 - 24x + 107

✓

Answer

f(x) = 2x³ - 24x + 107
f'(x) = 6x² - 24
= 6(x² - 4)
= 6(x + 2)(x - 2)
For f(x) to be increasing, we must have
f'(x) > 0
⇒ 6(x + 2)(x - 2) > 0
⇒ (x + 2)(x - 2) > 0
[Since, 6 > 0, 6(x + 2)(x - 2) > 0 ⇒ (x + 2)(x - 2) > 0]
⇒ x < -2 or x > 2
$\Rightarrow\text{x}\in(-\infty,-2)\cup(2,\infty)$
So, f(x) is increasing on $\text{x}\in(-\infty,-2)\cup(2,\infty).$
For f(x) to be decreasing, we must have,
f'(x) < 0
⇒ 6(x + 2)(x - 2) < 0
⇒ (x + 2)(x - 2) < 0
[Since, 6 > 0, 6(x + 2)(x - 2) < 0 ⇒ (x + 2)(x - 2) < 0]
⇒ -2 < x < 2
$\Rightarrow\text{x}\in(-2,2)$
So, f(x) is decreasing on $\text{x}\in(-2,2).$

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 the point on the curve x²= 8y which is nearest to the point (2,4).

Solve the following initial value problems:
$\Big\{\text{x}\sin^2\Big(\frac{\text{y}}{\text{x}}\Big)-\text{y}\Big\}\text{dx + x dy}=0,\text{y}(1)=\frac{\pi}4$

In a factory, machine A produces 30% of the total output, machine B produces 25% and the machine C produces the remaining output. If defective items produced by machines A, B and C are 1%, 1.2%, 2% respectively. Three machines working together produce 10000 items in a day. An item is drawn at random from a day's output and found to be defective. Find the probability that it was produced by machine B?

Find the equation of the plane which is perpendicular to the plane 5x + 3y + 6z + 8 = 0 and which contains the line of intersection of the planes x + 2y + 3z – 4 = 0 and 2x + y – z + 5 = 0.

Prove that the relation R on Z defined by $(\text{a, b})\in\text{R}\Leftrightarrow\ \text{a}-\text{b}$ is divisible by 5 is an equivalence relation on Z.

Using matrix method, solve the following system of equations:
$\frac{2}{\text{x}}+\frac{3}{\text{y}}+\frac{10}{\text{z}}=4,\frac{4}{\text{x}}-\frac{6}{\text{x}}+\frac{5}{\text{z}}=1,\frac{6}{\text{x}}+\frac{9}{\text{y}}-\frac{20}{\text{z}}=2; \text{ x,y,z,}\neq0$

Evaluate the following integrals:

$\int\frac{\cos\text{x}-\sin\text{x}}{\sqrt{8-\sin2\text{x}}}\text{ dx}$

Evaluate: $\int\frac{\text{x sin}^{-1}\text{x}}{{\sqrt{\text{x-1}^{2}}}}\text{dx}.$

Form the differential equation of the family of parabolas having vertex at origin and axis along positive y-axis.

Differentiate the following functions with respect to x:
$\log(\text{cosec x}-\cot\text{x})$