Class 12 Applications Of Derivatives · Very Long Answer5 Markslocal_and_absolute_extrema

Class 12 Maths · Applications Of Derivatives · Very Long Answer #65

For f(x)=sec x+ln(cos²x), 0<x<2π with x≠π/2,3π/2, classify all local extrema and decide whether absolute maximum and minimum values exist.

✅ Step-by-Step Solution

Final Answer: Local minima 2−2ln2 at π/3,5π/3; local maximum −1 at π; no absolute extrema.

The natural domain excludes cos x=0. Differentiate: f′=tan x(sec x−2).
Stationary inputs are π/3,π,5π/3. At cos x=1/2, f″=6>0; at x=π, f″=−3<0.
Therefore π/3 and 5π/3 are local minima, each with value 2+ln(1/4)=2−2ln2. At π the local maximum is −1.
As x approaches π/2 from the left, sec x→+∞ and dominates ln(cos²x), so f→+∞. From the right sec x→−∞ and f→−∞.
Hence no absolute maximum or minimum exists on the full domain; the stated finite values are local extrema.

[1 mark] The natural domain excludes cos x=0. Differentiate: f′=tan x(sec x−2).
[1 mark] Stationary inputs are π/3,π,5π/3. At cos x=1/2, f″=6>0; at x=π, f″=−3<0.
[1 mark] Therefore π/3 and 5π/3 are local minima, each with value 2+ln(1/4)=2−2ln2. At π the local maximum is −1.
[1 mark] As x approaches π/2 from the left, sec x→+∞ and dominates ln(cos²x), so f→+∞. From the right sec x→−∞ and f→−∞.
[1 mark] Hence no absolute maximum or minimum exists on the full domain; the stated finite values are local extrema.

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