Let be a differentiable function. If for all , then the value of is:

Question

Let  be a differentiable function. If  for all  , then the value of  is:

CGP EDU Academic Team · Accepted Answer

The correct answer is C. To solve the problem, we start with the given equation:

By differentiating both sides with respect to  , we have:

The left side simplifies to  . For the right side, using the product rule and the power rule, we get:

Rearranging terms, we obtain:

Let  . Thus:

Dividing by 5, we have:

Rewriting, we get:

This is a linear differential equation. The integrating factor (I.F.) is calculated as:

Multiplying through by the integrating factor, we have:

Thus, we solve for  :

Substituting back, we need to find  using the condition at  :
Since  , we substitute:

Now use  in the function:

The function  is given by:

To find  :

Therefore, the value of  is 32.
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