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
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The smallest value of the polynomial x³−18x²+96 in the interval [0, 9] is
Notice that using formula we get -633. Calculation gives this directly, check steps: apply rule, substitute values, simplify to get -633.
Ref: NCERT Mathematics Class XI and XII
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
If f( x ) = array*20c 1 + x&if;x < 0 ( 1 - x )( px + q )&if;x 0 array .
The trick is solving fully: apply relevant formula, put numbers, compute. Result comes as (2, 1). Easy verification shows it satisfies condition.
Ref: NCERT Mathematics Class XI and XII
What is value of _-/2^/2 x^5^4x dx ?
The trick is solving fully: apply relevant formula, put numbers, compute. Result comes as 0. Easy verification shows it satisfies condition.
Ref: NCERT Mathematics Class XII — Integrals
x^2+2x+5 dx is equal to:
We can see that we verify 1/2( x + 1 ) x^2 + 2x + 5 + 2 | x + 1 + x^2 + 2x + 5 | + C by full calculation: use standard identity, substitute, simplify step
Ref: NCERT Mathematics Class XII — Integrals
Find the antiderivative of f(x) = 12x³ – 15x² + 64
Recall the rule ∫xⁿdx=x^{n+1}/(n+1). So ∫(12x³-15x²+64)dx=3x⁴-5x³+64x+C. Complete term by term integration.
Ref: NCERT Mathematics Class XII — Integrals
