Let f (x) = x⁴ – 4x, then
Recall the rule solving fully: apply relevant formula, put numbers, compute. Result comes as f is increasing in [1,∞). Easy verification shows it satisfies condition.
Ref: NCERT Mathematics Class XI and XII
#derivative
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Let f(x) = x – [x], then f ‘ (x) = 1 for
The logic is straightforward: we verify all x ∈ (R−I) by full calculation: use standard identity, substitute, simplify step by step to reach all x ∈ (R−I).
Ref: NCERT Mathematics Class XI and XII
A particle is moving in a straight line and its distance x from a fixed point on the line at any time t seconds is given
Simply put, we verify 1 cm/sec by full calculation: use standard identity, substitute, simplify step by step to reach 1 cm/sec.
Ref: NCERT Mathematics Class XI — Coordinate Geometry
Given, f(x) = x³ – 12x² + 45x + 8. At which point does f(x) has its minimum?
Applying the basic concept, after simplifying, 5 is left. The method is simple: formula → substitution → calculation → answer.
Ref: NCERT Mathematics Class XI and XII
Given, f(x) = x³ – 12x² + 45x + 8. At which point does f(x) has its maximum?
In easy words, using formula we get 3. Calculation gives this directly, check steps: apply rule, substitute values, simplify to get 3.
Ref: NCERT Mathematics Class XI and XII
Given, f(x) = x³ – 12x² + 45x + 8. What is the minimum value of f(x)?
We can see that complete solution: write formula, plug data, do arithmetic. Final value matches 1. Simple and accurate.
Ref: NCERT Mathematics Class XII — Applications of Derivatives
Given, f(x) = x³ – 12x² + 45x + 8. What is the maximum value of f(x)?
We can see that we verify 62 by full calculation: use standard identity, substitute, simplify step by step to reach 62.
Ref: NCERT Mathematics Class XII — Applications of Derivatives
If y = enx, then nth derivative of y is:
Look at it this way: complete solution: write formula, plug data, do arithmetic. Final value matches nny. Simple and accurate.
Ref: NCERT Mathematics Class XI and XII
e^x f( x ) + f'( x ) dx; is equal to
We know that complete solution: write formula, plug data, do arithmetic. Final value matches e^xf( x ) +C. Simple and accurate.
Ref: NCERT Mathematics Class XII — Integrals
Let the slope of the curve y = sin-1 (cos x) be tan θ, then the value of θ in the interval (0, π) is
Understanding this is easy: complete solution: write formula, plug data, do arithmetic. Final value matches 3π/4. Simple and accurate.
Ref: NCERT Mathematics Class XI — Coordinate Geometry
What is the length of the longest interval in which the function f(x) = 3sin x – 4sin³ x is increasing?
From the definition, solving fully: apply relevant formula, put numbers, compute. Result comes as 3. Easy verification shows it satisfies condition.
Ref: NCERT Mathematics Class XII — Applications of Derivatives
The value of ε in the Lagrange's mean value theorem of f(b) - f(a) = (b - a) f'(ε) for f(x) = Ax² + Bx + C in (a, b) is
In easy words, using formula we get b + a/2. Calculation gives this directly, check steps: apply rule, substitute values, simplify to get b + a/2.
Ref: NCERT Mathematics Class XI — Sets, Relations and Functions
