Problem 44 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The solution set of \(f^{\prime}(x)>g^{\prime}(x)\) where \(f(x)=(1 / 2) 5^{2 x+1}\) and \(g(x)=5^{x}+4 x \log 5\) is (A) \((1, \infty)\) (B) \((0,1)\) (C) \((0, \infty)\) (D) \([0, \infty)\)

Step-by-Step Solution

Answer

The solution set is (0, ∞).

Step 1: Find the derivative of f(x)

To find the derivative of \( f(x) = \frac{1}{2} \cdot 5^{2x+1} \), use the chain rule. The derivative of \( 5^{2x+1} \) is \( 5^{2x+1} \cdot 2 \cdot \ln(5) \), so we have:\[f'(x) = \frac{1}{2} \cdot 5^{2x+1} \cdot 2 \ln(5) = 5^{2x+1} \ln(5)\]

Step 2: Find the derivative of g(x)

For \( g(x) = 5^x + 4x \ln(5) \), the derivative is computed by using the derivative of \( 5^x \) and \( x \ln(5) \) respectively:\[g'(x) = 5^x \ln(5) + 4 \ln(5)\]

Step 3: Set up the inequality

We need to solve the inequality \( f'(x) > g'(x) \), thus:\[5^{2x+1} \ln(5) > 5^x \ln(5) + 4 \ln(5)\]Cancelling \( \ln(5) \) from both sides, we get:\[5^{2x+1} > 5^x + 4\]

Step 4: Simplify the inequality

Rewrite \( 5^{2x+1} \) as \( 5 \cdot 5^{2x} \) to simplify:\[5 \cdot 5^{2x} > 5^x + 4\]Divide each term by \( 5^x \):\[5^{x+1} > 1 + \frac{4}{5^x}\]

Step 5: Analyze inequality behavior

As \( x \rightarrow \infty \), \( \frac{4}{5^x} \rightarrow 0 \), making \( 5^{x+1} \) always greater than 1 + small decreasing number. Test for boundary values if needed. It reveals \( x > 0 \) maintains the inequality valid, as at \( x = 0 \), it doesn't hold.

Key Concepts

Understanding Derivatives

A derivative measures how a function changes as its input changes. Essentially, it gives the slope of the function at any point. In simple terms, this means it tells us whether the function is increasing or decreasing, and how steeply.To illustrate, consider a straight road going uphill - the slope of the road tells you how steep the climb is. This is akin to a derivative, which indicates the rate of change of the function.When you find the derivative of a function, you symbolically express it with a prime, like in \( f'(x) \) for the function \( f(x) \). For example, the derivative of \( f(x) = \frac{1}{2} \, 5^{2x+1} \) is useful in determining the behavior of \( f'(x) \) in relation to another function's derivative, like that of \( g(x) \). Calculating the derivative is essential in solving calculus problems involving rates of change.

Applying the Chain Rule

The chain rule is a fundamental tool in calculus used to differentiate composite functions. Composite functions are those where one function is nested inside another. For example, when finding the derivative of \( 5^{2x+1} \), it isn't just a simple power of \( x \) but involves a function of \( x \) itself.With the chain rule, you differentiate the outer function and multiply it by the derivative of the inner function. Let’s break it down:

• Outer Function: \( y = 5^u \) where \( u = 2x+1 \)
• Derivative of Outer Function: \( dy/du = 5^u \ln(5) \)
• Inner Function: \( u = 2x+1 \)
• Derivative of Inner Function: \( du/dx = 2 \)

By applying the chain rule, the result is \( 5^{2x+1} \times 2 \times \ln(5) \), demonstrating the power of the chain rule in handling more complex derivatives.

Techniques of Inequality Solving

Solving inequalities is similar in spirit to solving algebraic equations, but with a couple of extra considerations since we deal with ranges of values rather than fixed solutions.When dealing with inequalities in calculus, especially those involving functions' derivatives, one usually requires the setup and simplification of the inequality. Take a scenario where finding the solution set for \( f'(x) > g'(x) \) is necessary:

• Start by expressing the derivatives, as shown in the initial exercise.
• Set up the inequality with these derivatives.
• Simplify this inequality by canceling common terms, which could involve logarithms or constants as seen in the solution \( \ln(5) \).
• Rearrange to get a more straightforward inequality. For instance, by expressing \( 5^{2x+1} \) to simplify to \( 5 \cdot 5^{2x} \).

After simplifying, it's crucial to analyze the behavior of the inequality as \( x \) approaches certain limits, such as infinity, to confirm if there are any critical points or ranges where the inequality holds. This dynamic approach ensures a better understanding of the behavior across different values of \( x \).