dont use graphs

Number of distinct real roots of f^''(x) if
f(x) is (x-1)²⁵(x-2)⁴³(x-3)¹¹ is ___

5 Answers

The solution has to do with graphs!!

without graphs i think it will be clumsy!

sir if i so i will edit the post sir..

i have got the graph for f(x) as

sir i have doubt in drawing the graph f^''(x)

you cant draw that directly (atleast i dont know!)

you have to find the points whwere f'(x) is zero..

actually now i have a feeeling that u should use the normal method only...

take f'x to be zero.. find the points.. (not as tough as u might feel) just a little dirty (messy) work

Then find f''x you have to find the points where f'x changes direction...

I think this combination should do..

√ √ || {} [] () → ~ ^ x x x → → ln log lg lim lim cos sin tan cot arcsin arccos arctan

∫ ∫ ∫ ∫ ∫ ∬ ∬ ∬ ∬ ∬ ∭ ∭ ∭ ∭ ∭ ⨌ ⨌ ⨌ ⨌ ⨌

∑ ∑ ∑ ∑ ∑ ∮ ∮ ∮ ∮ ∮ ∏ ∏ ∏ ∏ ∏ ∐ ∐ ∐ ∐ ∐

+ - × ÷ ± ∓ = ∪ ∩ ≠ ∼ ≈ ∅ ∀ ∃ ∈ ∋ ∝ ≤ ≥ ≃ ≅ ≡ < > ≪ ≫ ⊂ ∉ ⊃ ⊆ ⊇ ⇒ ⇐ ⇔ ⇒ ⇐ ⇔ ⇑ ⇓ ⇕ → ← ↔ → ← ↔ ↑ ↓ ↕ ↖ ↗ ↘ ↙ ↪ ↩ ⇀ ↼ ⇁ ↽

° ∂ ∞ α β γ δ θ λ μ ν ξ π ρ σ φ ω ε ƒ κ τ υ Γ Δ Λ Σ Π Ω