If the function f(x) = cases (1)/(|x|) & ,|x| ≥ 2 ax² + 2b & ,|x| < 2 cases$$f(x) = \begin{cases} \frac{1}{|x|} & ,|x| \geq 2 \\ ax^2 + 2b & ,|x| < 2 \end{cases}$$ is differentiable on R$\mathbb{R}$, then 48 (a + b)$48 (a + b)$ is equal to

Core Logic

The expression |x|$\cos|x|$ is everywhere differentiable. Thus, non-differentiability relies completely on the modulus function containing the quadratic factor, i.e., |x² - ax + 2|$|x^2 - ax + 2|$.

Non-differentiability points generally happen where:

Step 1: Evaluation of Roots

Given one of the roots is α = 2$\alpha = 2$:

Substituting a=3$a=3$ gives the other root β = 1$\beta = 1$. However, evaluating differentiability at x=1$x=1$ reveals properties that invalidate standard options.

Step 2: Conclusion

Due to a technical contradiction in the configuration of the differentiable constraints at x=1$x=1$, this question was officially dropped by NTA.

Pattern Recognition

If a quadratic inside a modulus has distinct real roots, it normally creates non-differentiable sharp turns unless a repeated multiplying factor outside cancels it out.

Chapter Mix

Class 12 Mathematics: Limits, Continuity and Differentiability

Related Formula

Standard indeterminacy layout resolution rules for limits matching form 1^∞$1^\infty$:

Core Logic

Evaluate structural integration base at boundary limit initialization state t → 0$t \to 0$:

This confirms it matches an indeterminate form of type 1^∞$1^\infty$.

Step 1: Apply Taylor Series or L'Hopital's Theorem

Compute limits of logarithmic integration properties inside exponential power indices:

Applying L'Hopital's theorem to differentiate the numerator with respect to t$t$ yields:

Evaluating this at t = 0$t = 0$ gives:

Step 2: Complete the Final Form Match

Integrating via parts results in logarithmic value updates:

Passing components back through exponential foundations transforms terms to:

Comparing with the target expression (α)/(5e)((8)/(5))(2)/(3)$\frac{\alpha}{5e}\left(\frac{8}{5}\right)^{\frac{2}{3}}$ isolates the numerical solution directly: α = 64$\alpha = 64$

Pattern Recognition

Treating 1^∞$1^\infty$ structural transformations using logarithmic derivatives allows managing complex functions containing multiple integral boundaries effectively.

Chapter Mix

Class 11 Mathematics: Limits Class 12 Mathematics: Definite Integration

Related Formula

A function is discontinuous if left-hand and right-hand limits mismatch at boundary transitions. Non-differentiability occurs at discontinuities or sharp turns.

Core Logic

Simplify the greatest integer component [x]$[x]$ by expanding over integer intervals:

Step 1: Testing Continuity Limits

Check continuity at structural boundaries: At x = 0$x = 0$: LHM = 0$\text{LHM} = 0$, RHM = 0$\text{RHM} = 0$ $\implies$ Continuous. At x = 1$x = 1$: LHM = 1$\text{LHM} = 1$, RHM = 3$\text{RHM} = 3$ $\implies$ Discontinuous. At x = 2$x = 2$: LHM = 4$\text{LHM} = 4$, RHM = 5$\text{RHM} = 5$ $\implies$ Discontinuous.

Thus, α = 2$\alpha = 2$ points of discontinuity (x in 1, 2$x \in \{1, 2\}$).

Step 2: Testing Differentiability Parameters

Discontinuities automatically introduce non-differentiability. Now check smooth corners at the remaining continuous transition x = 0$x = 0$:

Thus, β = 3$\beta = 3$ points of non-differentiability (x in 0, 1, 2$x \in \{0, 1, 2\}$).

Pattern Recognition

Discontinuities automatically break differentiability. Always count them first before checking derivatives at smooth corner points.

Chapter Mix

Class 12 Maths: Limits, Continuity and Differentiability

Related Formula

Continuity definition condition frame:

Core Logic

Evaluate Left-Hand Limit (LHL) using standard forms:

Given baseline definition states f(0) = 1+b$f(0) = 1+b$.

For Right-Hand Limit (RHL) to be finite and valid, the numerator tracking towards 0 means the denominator must also balance towards 0 to avoid divergence:

Step 1: Applying L'Hopital's rule to the RHL

With c=8$c=8$, evaluate RHL limit expressions using derivatives:

Step 2: Equating limits for parameter solutions

Equate continuous criteria milestones together: e^a = 1 + b = 3$e^a = 1 + b = 3$

Compute the ultimate target combination product configuration:

Pattern Recognition

Determining missing root constants inside indeterminate fraction structures handles calculations swiftly before running formal limits.

Chapter Mix

Class 12 Mathematics: Limits, Continuity and Differentiability

Related Formula

For a limit of the form x → 0 [g(x)]h(x)$\lim_{x \to 0} [g(x)]^{h(x)}$ where g(x) → 1$g(x) \to 1$ and h(x) → ∞$h(x) \to \infty$, the limit evaluates to:

Core Logic

Given the limit expression:

Using the standard form as f(2)=1$f(2)=1$, this transforms to:

Recognizing the definition of the derivative f'(2) = x → 0 (f(2+x)-1)/(x)$f'(2) = \lim_{x \to 0} \frac{f(2+x)-1}{x}$:

Given f'(2) = 4$f'(2) = 4$:

Step 1: Finding Intersection points with x-axis

Substitute α = 12$\alpha = 12$ into the equation of the curve:

To find where it meets the x-axis, set y = 0$y = 0$:

Testing for rational roots, x = -1$x = -1$ is a root because (-1)³ - (-1)² - 5(-1) - 3 = -1 - 1 + 5 - 3 = 0$(-1)^3 - (-1)^2 - 5(-1) - 3 = -1 - 1 + 5 - 3 = 0$.

Step 2: Factoring the cubic polynomial

Dividing x³ - x² - 5x - 3$x^3 - x^2 - 5x - 3$ by (x+1)$(x+1)$ gives:

The roots are x = -1$x = -1$ (repeated root) and x = 3$x = 3$. Therefore, the distinct real values of x$x$ where the curve intersects the x-axis are -1$-1$ and 3$3$, meaning it meets the x-axis exactly 2$2$ times.

Pattern Recognition

A repeated root like (x+1)²$(x+1)^2$ means the curve is tangent to the x-axis at that point, but it still counts as a meeting point. Always count distinct real roots when determining the number of meeting points with the coordinate axes.

Chapter Mix

Class 12 Mathematics: Limits, Continuity and Differentiability Class 11 Mathematics: Theory of Equations