Problem 13 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Let \(F(x)=f(x) g(x) h(x)\) for all real \(x\), where \(f(x), g\) \((x)\) and \(h(x)\) are differentiable functions. At some point \(x_{0}\), if \(F^{\prime}\left(x_{0}\right)=21 F\left(x_{0}\right), f^{\prime}\left(x_{0}\right)=4 f\left(x_{0}\right), g^{\prime}\left(x_{0}\right)=-7\) \(g\left(x_{0}\right)\) and \(h^{\prime}\left(x_{0}\right)=k h\left(x_{0}\right)\) then \(k\) is equal to (A) 24 (B) 12 (C) \(-12\) (D) \(-24\)

Step-by-Step Solution

Answer

The value of \( k \) is (A) 24.

Step 1: Differentiate the Given Function

Given the function \( F(x) = f(x)g(x)h(x) \), we use the product rule to differentiate. The derivative \( F'(x) \) is: \[ F'(x) = f'(x)g(x)h(x) + f(x)g'(x)h(x) + f(x)g(x)h'(x) \]

Step 2: Substitute Specific Values

Substitute \( x = x_0 \) into the derivative:\[ F'(x_0) = f'(x_0)g(x_0)h(x_0) + f(x_0)g'(x_0)h(x_0) + f(x_0)g(x_0)h'(x_0) \]We know that \( F'(x_0) = 21F(x_0) \), where \( F(x_0) = f(x_0)g(x_0)h(x_0) \).

Step 3: Express Derivatives in Terms of Given Ratios

Use the given data:- \( f'(x_0) = 4f(x_0) \)- \( g'(x_0) = -7g(x_0) \)- \( h'(x_0) = kh(x_0) \)Substitute these expressions into the derivative formula:\[ F'(x_0) = (4f(x_0)g(x_0)h(x_0)) + (f(x_0)(-7g(x_0))h(x_0)) + (f(x_0)g(x_0)kh(x_0)) \]

Step 4: Simplify the Expression

Simplify the expression:\[ F'(x_0) = 4f(x_0)g(x_0)h(x_0) - 7f(x_0)g(x_0)h(x_0) + kf(x_0)g(x_0)h(x_0) \]Combine terms:\[ F'(x_0) = (4 - 7 + k)f(x_0)g(x_0)h(x_0) \]This can be rewritten as:\[ F'(x_0) = (k - 3)f(x_0)g(x_0)h(x_0) \]

Step 5: Set Up the Equation to Solve for k

Since \( F'(x_0) = 21F(x_0) \), compare the two expressions:\[ (k - 3)f(x_0)g(x_0)h(x_0) = 21f(x_0)g(x_0)h(x_0) \]Cancel \( f(x_0)g(x_0)h(x_0) \) from both sides, provided it is non-zero:\[ k - 3 = 21 \]

Step 6: Solve for k

Solve the equation:\[ k - 3 = 21 \]\[ k = 21 + 3 \]\[ k = 24 \]

Key Concepts

Product Rule

When we are working with functions that are products of other functions, the product rule is a crucial tool in calculus. This rule helps in finding the derivative of a product of two or more functions. Suppose you have two functions, say \( u(x) \) and \( v(x) \). The product rule states that the derivative of the product \( u(x)v(x) \) is given by:

• \( (uv)' = u'v + uv' \)

For three functions \( f(x), g(x), \) and \( h(x) \) as in our exercise, the derivative is more involved:

• \( (fgh)' = f'gh + fg'h + fgh' \)

This accounts for each function being differentiated once while the others remain unchanged. Knowing this rule is essential for problems involving products of functions, like the one in our exercise.

Differentiation

Differentiation is a fundamental concept in calculus. It involves finding the derivative, which measures how a function changes as its input changes. In simple terms, a derivative can show both the rate of change and direction of a function.

In contexts like the exercise, differentiation uses rules such as the product rule to systematically break down complex expressions into manageable pieces. The derivative, in our scenario, captures the dynamics across multiple interacting functions: \( f(x), g(x), \) and \( h(x) \).

By applying rules such as the product rule and substituting known values, we convert theoretical knowledge into practical problem-solving steps. This approach is crucial when dealing with interconnected functions.

Derivatives

Derivatives express how a function changes at any given point. For example, the notation \( F'(x) \) represents the derivative of the function \( F(x) \). Derivatives are calculated using differentiation, as seen in our problem with the use of the product rule.

When we derived \( F'(x_0) \) involving three functions, we found

• \( F'(x_0) = f'(x_0)g(x_0)h(x_0) + f(x_0)g'(x_0)h(x_0) + f(x_0)g(x_0)h'(x_0) \)

In practical terms, solving for specific values like \( F'(x_0) \) means applying derivatives in conjunction with given conditions at a specific point \( x_0 \).

Derivatives are essential for analyzing and predicting the behavior of functions. They allow us to understand changes over small intervals and identify rates of change, culminating in clearer insights into mathematical models and real-world phenomena.