Find the intervals in which the function f given by f(x) = 2x3 − 3x2 − 36x + 7 is (a) strictly increasing (b) strictly decreasing
Solution and Explanation
The given function is f(x) = 2x3 − 3x2 − 36x + 7
\(f'(x)=6x^2-6x-36=6(x^2-x-6)=6(x+2)(x-3)\)
\(\therefore f'(x)=0\)⇒ x=-2, 3
The points x = −2 and x = 3 divide the real line into three disjoint intervals i.e.,
\((-\infin, -2),(-2,3)\) and \((3,\infin)\)
In interval \((-∞,-2)\) and \((3,∞)\), \(f'(x)\) is positive while in interval (-2,3), \(f'(x)\) is negative.
Hence, the given function (f) is strictly increasing in intervals \((-∞,-2)\) and \((3,∞)\), while function (f) is strictly decreasing in interval (−2, 3).
Top Questions on Applications of Derivatives
- Find the equation of the tangent to the curve \(y = x^2\) at the point (1,1).
- BITSAT - 2025
- Mathematics
- Applications of Derivatives
- Given that \( f(x) = \frac{\log x}{x} \), find the point of local maximum of \( f(x) \).
- CBSE CLASS XII - 2024
- Mathematics
- Applications of Derivatives
If \( x = a(0 - \sin \theta) \), \( y = a(1 + \cos \theta) \), find \[ \frac{dy}{dx}. \]
- UP Board XII - 2024
- Mathematics
- Applications of Derivatives
Find the least value of ‘a’ for which the function \( f(x) = x^2 + ax + 1 \) is increasing on the interval \( [1, 2] \).
- UP Board XII - 2024
- Mathematics
- Applications of Derivatives
- Ahousing society wants to commission a swimming pool for its residents. For this, they have to purchase a square piece of land and dig this to such a depth that its capacity is 250 cubic metres. Cost of land is 500 per square metre. The cost of digging increases with the depth and cost for the whole pool is 4000 (depth)2. Suppose the side of the square plot is x metres and depth is h metres.
On the basis of the above information, answer the following questions
Given:- Square plot side = x metres
- Depth = h metres
- Volume = x2h = 250 cubic metres
- Cost of land: 500 per sq.metre
- Cost of digging: 4000 ×h- CBSE CLASS XII - 2023
- CBSE Compartment XII - 2023
- Mathematics
- Applications of Derivatives
Questions Asked in CBSE CLASS XII exam
- What is ‘Misch metal’? Give its one use.
- CBSE CLASS XII - 2025
- Inorganic chemistry
- Using integration, find the area of the region bounded by the line \[ y = 5x + 2, \] the \( x \)-axis, and the ordinates \( x = -2 \) and \( x = 2 \).
- CBSE CLASS XII - 2025
- Evaluation of Definite Integrals by Substitution
- Your school is organizing an Inter-House Science Model-Making Competition. As President of the Science Club, draft a notice to inform all House members from IX – XII about the competition and specify the number of registrations invited per house. Include other necessary details. You are Mitali/Mukesh. Put your notice in a box.
- CBSE CLASS XII - 2025
- Letter Writing
- Which one of the following is the main polluter of Yamuna water in Delhi?
- CBSE CLASS XII - 2025
- Water pollution
- The objective of state highways is:
- CBSE CLASS XII - 2025
- Transport and Communication
Concepts Used:
Increasing and Decreasing Functions
Increasing Function:
On an interval I, a function f(x) is said to be increasing, if for any two numbers x and y in I such that x < y,
⇒ f(x) ≤ f(y)
Decreasing Function:
On an interval I, a function f(x) is said to be decreasing, if for any two numbers x and y in I such that x < y,
⇒ f(x) ≥ f(y)
Strictly Increasing Function:
On an interval I, a function f(x) is said to be strictly increasing, if for any two numbers x and y in I such that x < y,
⇒ f(x) < f(y)
Strictly Decreasing Function:
On an interval I, a function f(x) is said to be strictly decreasing, if for any two numbers x and y in I such that x < y,
⇒ f(x) > f(y)
Graphical Representation of Increasing and Decreasing Functions
