34 Let f :[1,3] arrow[0, ∞) be continuous and differentiable function and if (f(3)-f(1)) ⋅(f2(3)+f2(1)+f(3) f(1))=k f2(c) f prime(c) where c ∈(1,3), then find the value of k.

Question

34 Let f :[1,3] arrow[0, ∞) be continuous and differentiable function and if (f(3)-f(1)) ⋅(f2(3)+f2(1)+f(3) f(1))=k f2(c) f prime(c) where c ∈(1,3), then find the value of k. (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

Let F(x)=f3(x) and F(x) is continuous and differentiable function in [1,3]. ∴ (F(3)-F(1)/3-1)=F prime(c) text (using L.M.V.T.) (f3(3)-f3(1)/2)=3 f2(c) ⋅ f prime(c) ⇒ k=6

Q. 34 Let $f : [1, 3] \to [0, \infty)$ be continuous and differentiable function and if $(f (3) - f (1)) \cdot (f^{2} (3) + f^{2} (1) + f (3) f (1)) = k f^{2} (c) f^{'} (c)$ where $c \in (1, 3)$ , then find the value of $k$ .

Let $F (x) = f^{3} (x)$
and $F (x)$ is continuous and differentiable function in $[1, 3]$ .
$∴ \frac{F ( 3 ) - F ( 1 )}{3 - 1} = F^{'} (c) (using L.M.V.T.)$
$\frac{f ^{3} ( 3 ) - f ^{3} ( 1 )}{2} = 3 f^{2} (c) \cdot f^{'} (c) \Rightarrow k = 6$

Q. 34 Let f:[1,3]→[0,∞) be continuous and differentiable function and if (f(3)−f(1))⋅(f2(3)+f2(1)+f(3)f(1))=kf2(c)f′(c) where c∈(1,3), then find the value of k.

Let F(x)=f3(x) and F(x) is continuous and differentiable function in [1,3]. ∴3−1F(3)−F(1)​=F′(c) (using L.M.V.T.) 2f3(3)−f3(1)​=3f2(c)⋅f′(c)⇒k=6

Q. 34 Let $f : [1, 3] \to [0, \infty)$ be continuous and differentiable function and if $(f (3) - f (1)) \cdot (f^{2} (3) + f^{2} (1) + f (3) f (1)) = k f^{2} (c) f^{'} (c)$ where $c \in (1, 3)$ , then find the value of $k$ .

Let $F (x) = f^{3} (x)$
and $F (x)$ is continuous and differentiable function in $[1, 3]$ .
$∴ \frac{F ( 3 ) - F ( 1 )}{3 - 1} = F^{'} (c) (using L.M.V.T.)$
$\frac{f ^{3} ( 3 ) - f ^{3} ( 1 )}{2} = 3 f^{2} (c) \cdot f^{'} (c) \Rightarrow k = 6$