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Applications Of Derivatives - II — Flashcards

ICSE · Class 12 · Mathematics

29 flashcards for Applications Of Derivatives - II (ICSE Class 12 Mathematics) to test yourself on key terms and facts.

52 questions29 flashcards10 formulas & key relations5 concepts

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A graph of a continuous function on a closed interval [a, b], clearly indicating the absolute maximum and absolute minimum values and their corresponding points.
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29 Flashcards·
Absolute extrema on the real lineAbsolute extrema on a closed intervalLocal extrema and end pointsCritical pointsFirst derivative testSecond derivative test
Card 1Absolute extrema on the real line

Solve for the absolute maximum and absolute minimum of f(x)=|x| on R.

Answer

Step 1: For every real x, |x| >= 0. Step 2: The smallest value is 0, and it occurs at x = 0. Step 3: There is no largest value because |x| keeps growing as |x| becomes larger. Answer: Absolute minimum…

Card 2Absolute extrema on a closed interval

Find the absolute maximum and absolute minimum of f(x)=|x| on [2,5].

Answer

Step 1: On [2,5], x is positive, so |x| = x. Step 2: The smallest value on the interval is at x = 2. Step 3: The largest value on the interval is at x = 5. Step 4: f(2)=2 and f(5)=5. Answer: Absolute …

Card 3Local extrema and end points

Why can a local maximum or local minimum not occur at an end point of an interval?

Answer

A local extremum needs a whole neighbourhood around the point. At an end point, there is no open interval on both sides inside the domain, so the neighbourhood condition fails. Example: an end point c…

Card 4Critical points

Find the critical points of f(x)=x^3+x^2+x+1.

Answer

Step 1: Differentiate: f'(x)=3x^2+2x+1. Step 2: Set f'(x)=0: 3x^2+2x+1=0. Step 3: The discriminant is 2^2-4(3)(1)=4-12=-8, so there are no real roots. Step 4: f'(x) does not fail to exist anywhere. An…

Card 5First derivative test

Solve: Find the local maximum or minimum of f(x)=2x^3-6x^2+6x+5 using the first derivative test.

Answer

Step 1: Differentiate: f'(x)=6x^2-12x+6=6(x-1)^2. Step 2: Critical point: f'(x)=0 gives x=1. Step 3: Check sign near x=1. Since 6(x-1)^2 is positive on both sides of 1, the sign does not change. Step …

Card 6First derivative test

Find the local maximum and local minimum of f(x)=(x-3)^4 using the first derivative test.

Answer

Step 1: f'(x)=4(x-3)^3. Step 2: Set f'(x)=0 gives x=3. Step 3: For x slightly less than 3, f'(x)<0. For x slightly more than 3, f'(x)>0. Step 4: Negative to positive means local minimum. Step 5: f(3)=…

Card 7First derivative test

Find the local maximum of f(x)=x*sqrt(1-x) on (0,1).

Answer

Step 1: Differentiate. f'(x)=x*(-1/(2sqrt(1-x)))+sqrt(1-x)=(2-3x)/(2sqrt(1-x)). Step 2: Set numerator equal to zero: 2-3x=0, so x=2/3. Step 3: For x slightly less than 2/3, f'(x)>0. Step 4: For x slig…

Card 8Second derivative test

Use the second derivative test for f(x)=2x^3-21x^2+36x-20.

Answer

Step 1: f'(x)=6x^2-42x+36=6(x-1)(x-6). Step 2: Critical points are x=1 and x=6. Step 3: f''(x)=12x-42. Step 4: f''(1)= -30 < 0, so x=1 is a local maximum. Step 5: f(1)=2-21+36-20=-3. Step 6: f''(6)=30…

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Frequently Asked Questions

What are the important topics in Applications Of Derivatives - II for ICSE Class 12 Mathematics?
Key topics in Applications Of Derivatives - II include Absolute Maximum, Absolute Minimum, and Absolute Extremum, Critical Points, Stationary Points, and Inflexion, First Derivative Test, Second Derivative Test. Study these first, then practise questions on each for the ICSE Class 12 board exam.
How many flashcards are available for Applications Of Derivatives - II?
There are 29 flashcards for Applications Of Derivatives - II covering key definitions, facts and ideas. A few sample cards are shown on this page.
How should I revise Applications Of Derivatives - II for the ICSE Class 12 board exam?
Learn the core ideas first, then work through the 52 practice questions on Applications Of Derivatives - II. Revise definitions regularly and use flashcards for quick recall before the exam.

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