Question

f(x)=x^3-4x on [-3,0] Verify Rolle's Theorem

Answer 1

f(x) = x (x + 3)e–x/2 is continuous in [–3, 0] and f′(x)=(x2 + 3x) (– 1/2)e-x/2 + (2x + 3)e–x/2 = – 1/2(x2 – x – 6)e–x/2 Therefore f′(x) exists (i.e., finite) for all x Also f(–3) = 0, f (0) = 0 So that f(–3) = f(0) Hence all the three conditions of the theorem are satisfied. Now consider f′(c)=0 i.e., –1/2(c2 – c – 6)e–c/2 = 0 c2 – c – 6 = 0 (c + 2) (c – 3) = 0 c = 3 or –2 Hence there exists – 2 ∈ (–3, 0) such that f′(–2) = 0 and hence Rolle’s theorem is verified

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