Given f2(x)+g2(x)+h2(x) ≤ 9 and U(x)=3 f(x)+4 g(x) +10 h(x), where f(x), g(x) and h(x) are continuous ∀ x ∈ R. If maximum value of U(x) is √N, then find N.

Question

Given f2(x)+g2(x)+h2(x) ≤ 9 and U(x)=3 f(x)+4 g(x) +10 h(x), where f(x), g(x) and h(x) are continuous ∀ x ∈ R. If maximum value of U(x) is √N, then find N. (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

Let vecV1=3 hati+4 hatj+10 hatk and vecV2=f(x) hati+g(x) hatj+h(x) hatk U(x)= vecV1 ⋅ vecV2=| vecV1|| vecV2| cos θ ≤| vecV1|| vecV2| =√9+16+100 √f2+g2+h2 ≤ 3 √125=√1125 ∴ N=1125

Q. Given f2(x)+g2(x)+h2(x)≤9 and U(x)=3f(x)+4g(x) +10h(x), where f(x),g(x) and h(x) are continuous ∀x ∈R. If maximum value of U(x) is N​, then find N.

Let V1​=3i^+4j^​+10k^ and V2​=f(x)i^+g(x)j^​+h(x)k^ U(x)=V1​⋅V2​=∣∣​V1​∣∣​∣∣​V2​∣∣​cosθ ≤∣∣​V1​∣∣​∣∣​V2​∣∣​ =9+16+100​f2+g2+h2​≤3125​=1125​ ∴N=1125

Q. Given $f^{2} (x) + g^{2} (x) + h^{2} (x) \leq 9$ and $U (x) = 3 f (x) + 4 g (x)$ $+ 10 h (x)$ , where $f (x), g (x)$ and $h (x)$ are continuous $\forall x$ $\in R$ . If maximum value of $U (x)$ is $N$ , then find $N$ .