Question

If $f:\mathbb{R}\to \mathbb{R}$ is a differentiable function such that $f^{\prime }(x)>2f(x)$ for all $x\in \mathbb{R}$, and $f(0)=1$, then [A] $f(x)$ is increasing in $(0,\mathrm{\infty })$ [B] $f(x)$ is decreasing in $(0,\mathrm{\infty })$ [C] $f(x)>e^{2x}$ in $(0,\mathrm{\infty })$ [D] \(f^{\prime}(x)

Short answer

Option [A] is correct since $f^{\prime }(x)>0$ implies that the function is increasing. The veracity of options [C] and [D] cannot be determined without additional proof that $f(x)>e^{2x}$ for all $x>0$.

Step-by-step solution

Understanding the Given Information

We know that the function is differentiable, which means that it's slope (or rate of change), represented by its derivative, is continuous. We're given that the derivative of the function at any point is greater than twice the function's value at that point, expressed as $f^{\prime }(x)>2f(x)$. Moreover, the initial condition is $f(0)=1$.

Analyzing the Initial Condition

The condition $f(0)=1$ tells us the value of the function at $x=0$. It helps us to evaluate the function and its growth pattern over the interval $(0,\mathrm{\infty })$.

Examining Option [A] - Increassing Nature of f(x)

If $f^{\prime }(x)>2f(x)$ and considering that $f(x)>0$ since $f(0)=1$, it implies that $f^{\prime }(x)>0$ for all $x$. A positive derivative indicates that the function is increasing. Therefore, option [A] is correct.

Eliminating Option [B] - Decreasing Nature of f(x)

From the analysis of [A], since $f^{\prime }(x)>0$, $f(x)$ cannot be decreasing in $(0,\mathrm{\infty })$. Hence, option [B] is incorrect.

Analyzing Option [C] - Comparing f(x) and $e^{2x}$

Consider the function $g(x)=f(x)-e^{2x}$. Now, $g^{\prime }(x)=f^{\prime }(x)-(e^{2x}{)}^{\prime }=f^{\prime }(x)-2e^{2x}$ and we know that $f^{\prime }(x)>2f(x)$ and $f(0)=1>e^0=1$. Therefore, for very small positive values of $x$, $f^{\prime }(x)>2f(x)>2e^{2x}$, which implies $g^{\prime }(x)>0$. Now, this indicates that $g(x)$ is initially increasing, and since $g(0)>0$, it suggests $g(x)$ is likely to remain positive and hence, $f(x)>e^{2x}$ in $(0,\mathrm{\infty })$ if $g(x)$ never crosses the x-axis. However, to definitely confirm this, one would typically need to verify that $g(x)$ does not have any zeros for $x>0$. Without this proof, we cannot confirm option [C].

Verifying Option [D] - Comparison of f'(x) and $e^{2x}$

Given that $f^{\prime }(x)>2f(x)$, and if we incorrectly assume $f(x)>e^{2x}$, we might be tempted to say $f^{\prime }(x)>2e^{2x}$, which would imply option [D] is incorrect. However, we have not confirmed $f(x)>e^{2x}$ for all $x>0$, so we cannot reliably evaluate option [D] without further information.

Key concepts

Differentiable Functions

Understanding differentiable functions is crucial in advanced mathematics, particularly in calculus. A function is said to be differentiable at a point when its derivative exists at that point. In simple terms, if you can determine the exact slope of the tangent line to the function at any point, then the function is differentiable there. This property of differentiability implies continuity; meaning not only does the function have a value at that point, but it also behaves nicely, without any sharp turns or jumps.

For the JEE Advanced Mathematics context, when dealing with differentiable functions like in the exercise provided, you can infer several properties about the function's behavior. Given the fact that a differentiable function has a continuous rate of change, we can analyze the function's growth by looking at the sign of its derivative. For example, if the derivative, denoted by $f^{\prime }(x)$, is positive across an interval, we can conclude that the function is increasing over that same interval. In our exercise, the function's derivative isn't just positive; it's greater than twice the function's value, indicated by the inequality $f^{\prime }(x)>2f(x)$, which guarantees a certain rapidity in growth.

Function Growth

The growth of a function refers to how its value either increases or decreases as the input, commonly the $x$-value, changes. In the context of calculus and the subject exercise, we're particularly looking at how the value of $f(x)$ changes. When we're told that the derivative $f^{\prime }(x)$ is positive, it points to the function increasing. This means as $x$ gets larger, so does $f(x)$, moving upwards on a graph.

This idea becomes even more interesting when the derivative surpasses a certain multiple of the function itself. With $f^{\prime }(x)>2f(x)$, we're dealing with exponential-type growth - this isn't just a gentle incline, it's a steep one. This information tells us more than just the increase; it suggests the function is likely to outstrip other functions that grow at a slower rate, such as linear functions, over the same interval. If we leverage this concept, we can also compare the growth rate of our function to other well-known functions, like the exponential function $e^{2x}$, which plays a key role in analyzing the function's behavior in parts [C] and [D] of the given exercise.

Mathematical Inequalities

Mathematical inequalities are comparisons between two values or expressions, indicating that one is larger or smaller than the other. They are fundamental in understanding function behavior and setting up conditions in problems like the one presented. Inequalities such as $f^{\prime }(x)>2f(x)$ in the exercise enable us to deduce a variety of conclusions about the function's nature over an interval.

For instance, knowing that $f^{\prime }(x)>2f(x)$ gives us an inequality that defines how steeply $f(x)$ is rising: faster than double itself at any point. This has profound implications not only on the increasing nature of the function but also on how we compare it to other functions—such as the exponential function in the exercise. Moreover, with the initial condition $f(0)=1$, we understand that the function has a specific starting point. We use these inequalities to predict or prove certain behaviors, like in the process of deducing whether $f(x)>e^{2x}$ over a range of values; this involves understanding the inequality deeply, within the context of other mathematical principles like limits and asymptotic behavior.