Let g(x) be a continuous and differentiable function such that ∫02 ∫25/2 2x2−3dx⋅g(x)dx=0 then g(x) = 0 when x∈(0,2) has: (where [.] denote greatest integer function)

As 1 0 ∀x∈(0,2)⇒g(x)=0 should have at least one real root in (0, 2) .

Let g(x) be a continuous and differentiable function such that ∫02 ∫25/2 2x2−3dx⋅g(x)dx=0 then g(x) = 0 when x∈(0,2) has: (where [.] denote greatest integer function)

As 1 0 ∀x∈(0,2)⇒g(x)=0 should have at least one real root in (0, 2) .

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Let g(x) be a continuous and differentiable function such that $\int_{0}^{2} (\int_{\sqrt{2}}^{\sqrt{5 / 2}} [2 x^{2} - 3] dx) \cdot g (x) dx = 0$ then g(x) = 0 when $x \in (0, 2)$ has: (where [.] denote greatest integer function)

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Exactly one real root

Atleast one real root

No real root

Two real roots

answer is B.

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As $1 < 2 x^{2} - 3 < 2$ $\forall x \in (\sqrt{2}, \sqrt{\frac{5}{2}})$

$∴ \int_{\sqrt{2}}^{\sqrt{5 / 2}} [2 x^{2} - 3] dx > 0 \forall x \in (0, 2)$

$\Rightarrow g (x) = 0$ should have at least one real root in (0, 2) .

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Let g(x) be a continuous and differentiable function such that ∫02 ∫25/2 2x2−3dx⋅g(x)dx=0 then g(x) = 0 when x∈(0,2) has: (where [.] denote greatest integer function)

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Let g(x) be a continuous and differentiable function such that $\int_{0}^{2} (\int_{\sqrt{2}}^{\sqrt{5 / 2}} [2 x^{2} - 3] dx) \cdot g (x) dx = 0$ then g(x) = 0 when $x \in (0, 2)$ has: (where [.] denote greatest integer function)