To determine the points at which the function \( f(x) = \max\{6x, 2 + 3x^2\} + |x - 1| \cos|x^2 - \frac{1}{4}| \) is not differentiable, we need to consider the differentiability of both components: the max function and the absolute value function.
The function \( g(x) = \max\{6x, 2 + 3x^2\} \) is not differentiable where the expressions inside the max are equal, i.e., \( 6x = 2 + 3x^2 \). This simplifies to:
\( 3x^2 - 6x + 2 = 0 \).
Using the quadratic formula, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), with \( a = 3, b = -6, c = 2 \), gives:
\( x = \frac{6 \pm \sqrt{36 - 24}}{6} = \frac{6 \pm \sqrt{12}}{6} = \frac{6 \pm 2\sqrt{3}}{6} = 1 \pm \frac{\sqrt{3}}{3} \).
These are the points where \( g(x) \) is not differentiable.
Next, consider \( h(x) = |x - 1| \cos|x^2 - \frac{1}{4}| \), which is not differentiable where \( x - 1 = 0 \), i.e., \( x = 1 \).
Now, combine these potential non-differentiable points:
| \(1 + \frac{\sqrt{3}}{3}\) |
| \(1 - \frac{\sqrt{3}}{3}\) |
| 1 |
All points lie within \( x \in (-\pi, \pi) \).
Thus, the number of points where \( f(x) \) is not differentiable is 3.
The result, 3, verifies the given range 3,3.
Similar Questions
- If $f(x)$ satisfies the relation $f(x) = e^x + \int_0^1 (y + xe^x)f(y)dy$, then $e + f(0)$ is equal to _______.
- $6 \int_0^\pi |(\sin 3x + \sin 2x + \sin x)| dx$ is equal to ____________.
- If $\int_0^1 4 \cot^{-1}(1 - 2x + 4x^2) dx = a \tan^{-1}(2) - b \log_e(5)$, where $a, b \in \mathbf{N}$, then $(2a + b)$ is equal to __________.
- Let the area of the region bounded by the curve $y = \max\{\sin x, \cos x\}$, lines $x = 0$, $x = \frac{3\pi}{2}$, and the $x$-axis be $A$. Then, $A + A^2$ is equal to ___________.
- The number of elements in the set $S = \left\{ x : x \in [0, 100] \text{ and } \int_{0}^{x} t^2 \sin(x-t)dt = x^2 \right\}$ is _____
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Let $f(x) = \begin{cases} e^{x-1} & , x < 0 \\ x^2 - 5x + 6 & , x \geq 0 \end{cases}$ and $g(x) = f(|x|) + |f(x)|$. If the number of points where $g$ is not continuous and is not differentiable are $\alpha$ and $\beta$ respectively, then $\alpha + \beta$ is equal to ________
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Important Questions from Calculus
- If $f(x)$ satisfies the relation $f(x) = e^x + \int_0^1 (y + xe^x)f(y)dy$, then $e + f(0)$ is equal to _______.
- $6 \int_0^\pi |(\sin 3x + \sin 2x + \sin x)| dx$ is equal to ____________.
- If $\int_0^1 4 \cot^{-1}(1 - 2x + 4x^2) dx = a \tan^{-1}(2) - b \log_e(5)$, where $a, b \in \mathbf{N}$, then $(2a + b)$ is equal to __________.
- Let the area of the region bounded by the curve $y = \max\{\sin x, \cos x\}$, lines $x = 0$, $x = \frac{3\pi}{2}$, and the $x$-axis be $A$. Then, $A + A^2$ is equal to ___________.
- The number of elements in the set $S = \left\{ x : x \in [0, 100] \text{ and } \int_{0}^{x} t^2 \sin(x-t)dt = x^2 \right\}$ is _____
