FunctionHard

Question

Let f(x) is a continuous and twice differentiable, even function in R. If f(4) = - 4, f(-2) = 4, f(0) = - 4 f(x) = 8 = f(x), then minimum number of solutions of equation f″(x) + 2f′(x) + f(x) = 0 will be -

Options

A.2
B.3
C.4
D.5

Solution

           
Let φ(x) = f(x)ex
φ′(x) = (f(x) + f′(x))ex
φ″(x) = (f″(x) + 2f′(x) + f(x))ex
There will be atleast one root of f(x) in each of the following interval (-∞, -4), (-4, -2), (-2, 0) (0, 2), (2, 4), (4, ∞)
so f(x) will have at least 6 roots
∴ φ″ f″(x) = 0 will have at least 4 roots.

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