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f(x) = a.(b x c) with x in a, b and c: value of a.b at the point of local minimum

The same question, asked 2 times: 2 May 2023 S1*, 14 May 2024 S2. Every wording below, oldest first.

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  1. Question 1

    2 May 2023 S1

    126
    If x0x_0 is the point of local minima of f(x)=a⃗⋅(b⃗×c⃗)f(x) = \vec{a}\cdot(\vec{b}\times\vec{c}) where a⃗=xi^−2j^+3k^\vec{a} = x\hat{i} - 2\hat{j} + 3\hat{k}, b⃗=−2i^+xj^−k^\vec{b} = -2\hat{i} + x\hat{j} - \hat{k}, c⃗=7i^−2j^+xk^\vec{c} = 7\hat{i} - 2\hat{j} + x\hat{k}, then value of a⃗⋅b⃗\vec{a}\cdot\vec{b} at x=x0x = x_0 is

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  2. Question 2

    14 May 2024 S2

    127
    Let x0x_0 be the point of local minima of f(x)=a⃗⋅(b⃗×c⃗)f(x) = \vec{a}\cdot(\vec{b}\times\vec{c}) where a⃗=xi^−2j^+3k^\vec{a} = x\hat{i}-2\hat{j}+3\hat{k}, b⃗=−2i^+xj^−k^\vec{b} = -2\hat{i}+x\hat{j}-\hat{k}, c⃗=7i^−2j^+xk^\vec{c} = 7\hat{i}-2\hat{j}+x\hat{k}, then value of a⃗⋅b⃗\vec{a}\cdot\vec{b} at x=x0x = x_0 is

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