Question

Let, F : R --> R be a thrice differentiable function.

Suppose that,

F(1) = 0, F(3)= -4 and F'(x)< 0 for all
x € (1/2,3).

Let, f(x) = x F(x) for all x € R.

The correct statement(s) is(are) :-

(a) f'(1)< 0

(b) f(2) < 0

(c) f'(x) ≠ 0 for any x € (1,3)

(d) f'(x) = 0 for some x € (1,3)


✔️✔️ Proper solution needed ✔️✔️​

Answer 1

Answer:

Step-by-step explanation:

brainliest please

Answer 2

ANSWER

(a) f'(1)< 0

(b) f(2) < 0

(c) f'(x) ≠ 0 for any x € (1,3)

These are correct options.

Step By step Explanation

We have >

F(3) = -4.

F(1) = 0.

F(x) = x F(x)

f(1) = 1 F(1) = 0 =>f(2) < 0

f(3) = 1 F(3) = -12

Now,

f'(x) = F(x) + x F'(x)

=> F(x)<0, F'(x)<0

=> f'(x) <0 for x->(1,3)

f'(c) = ${f(3) - f(1)}{3-1}$

=> f'(c) = $\dfrac{-12-0}{2}$

=> f'(c) = -6

$C \epsilon (1,3)$

Also,

f'(1) = F(1) + F'(1)

=> f'(1) = 0 + F'(1)

=> F'(1) < 0.

Hence we get that option a,b and c are correct here.