Animated Solution for Mathematics - Limits, Continuity and Differentiability: If the function $f(x)=\{\begin{array}{ll}\frac{1}{|x|}& ,|x|\ge 2\\ a{x}^2+2b& ,|x|<2\end{array}$ is differentiable on $\mathbb{R}$, then $48(a+b)$ is equal to_____

Imagine you are standing on a path that suddenly changes its nature. One moment, you are walking along a smooth, elegant curve defined by $f(x)=\frac{1}{|x|}$, and the next, you are transitioning onto a parabolic path defined by $f(x)=a{x}^2+2b$.

In the world of calculus, we call this a piecewise function. The question before us is not just about finding values for $a$ and $b$; it is about ensuring that this transition is perfectly smooth. We want to ensure that as you move from the hyperbola to the parabola, there is no sudden jump and no sharp, jarring corner.

Continuity

Before we can talk about smoothness, we must talk about connection. A function cannot be differentiable at a point if it is not even continuous there. If there were a gap at $x=2$, the curve would be broken, and the concept of a tangent line would lose its meaning.

Our first task is to ensure the two branches meet at the same $y$-coordinate at $x=2$. For the right branch, as $x$ approaches $2$, $f(x)=\frac{1}{|x|}$ becomes:

For the left branch, the parabola $f(x)=a{x}^2+2b$ must also yield $\frac{1}{2}$ at $x=2$. Substituting $x=2$ into the parabola, we get $4a+2b$. Equating these, we arrive at our first vital equation:

The Slope

Now that we have ensured the paths meet, we must ensure they merge without a sharp turn. This is where the derivative comes in, as it represents the slope of the tangent line.

For the transition to be smooth, the slope of the hyperbola as it approaches $x=2$ must be identical to the slope of the parabola at that same point. Let us calculate the slope of the right branch:

Now, let us calculate the slope of the left branch. The derivative of $a{x}^2+2b$ is $2ax$. At $x=2$, this slope is:

For the transition to be perfectly smooth, we set these slopes equal:

Now, we return to our continuity equation: $4a+2b=\frac{1}{2}$. Substituting our value $a=-\frac{1}{16}$, we get:

Adding $\frac{1}{4}$ to both sides, we get $2b=\frac{3}{4}$, which means $b=\frac{3}{8}$. We have now found both $a$ and $b$.

The final step is to calculate $48(a+b)$. Adding $a$ and $b$ gives us:

Finally, we compute the result:

Through the rigorous application of continuity and differentiability, we have tamed this piecewise function. The final answer is 15.