The number of critical points of the function $f(x) = (x - 2)^{2/3}(2x + 1)$ is:
- 2
- 0
- 1
- 3
The Correct Option is A
Approach Solution - 1
To find the number of critical points of the function \(f(x) = (x - 2)^{2/3}(2x + 1)\), we need to determine the points where the derivative \(f'(x)\) is either zero or undefined.
Step 1: Find the derivative \(f'(x)\).
The product rule is applicable here since \(f(x)\) is a product of two functions: \(g(x) = (x - 2)^{2/3}\) and \(h(x) = (2x + 1)\).
The derivative is given by:
\(f'(x) = g'(x)h(x) + g(x)h'(x)\)
First, find \(g'(x)\) where \(g(x) = (x - 2)^{2/3}\):
\(g'(x) = \frac{2}{3}(x - 2)^{-1/3}\)
Next, find \(h'(x)\) where \(h(x) = 2x + 1\):
\(h'(x) = 2\)
Substituting these into the product rule formula:
\({f'(x) = \frac{2}{3}(x - 2)^{-1/3}(2x + 1) + (x - 2)^{2/3} \cdot 2}\)
Simplifying the expression, we have:
\(f'(x) = \frac{2(2x + 1)}{3(x - 2)^{1/3}} + 2(x - 2)^{2/3}\)
Step 2: Determine where \(f'(x) = 0\) or where \(f'(x)\) is undefined.
The expression is undefined at \(x = 2\) because \((x - 2)^{-1/3}\) becomes undefined.
Next, set \(f'(x) = 0\) and solve for \(x\):
\(\frac{2(2x + 1)}{3(x - 2)^{1/3}} + 2(x - 2)^{2/3} = 0\)
Factor out common terms and simplify to find the possible roots (you may need to equate numerators if possible).
After simplification, the critical points arise at \(x = 1\) (from solving \(f'(x) = 0\)) and \(x = 2\) (as \(f'(x)\) is undefined).
Conclusion: The function \(f(x) = (x - 2)^{2/3}(2x + 1)\) has two critical points: one where the derivative is zero and one where it is undefined.
Thus, the number of critical points is 2.
Approach Solution -2
Solution:
The given function is \(f(x)\). Its derivative is:
\[ f'(x) = \frac{2}{3}(x - 2)^{-1/3}(2x + 1) + (x - 2)^{2/3}(2). \]
Simplify the numerator:
\[ f'(x) = \frac{2}{3} \cdot \frac{(2x + 1) + 3(x - 2)}{(x - 2)^{1/3}}. \]
Expand and simplify:
\[ (2x + 1) + 3(x - 2) = 5x - 5. \]
Thus:
\[ f'(x) = \frac{2(5x - 5)}{3(x - 2)^{1/3}}. \]
Critical points:
- Set \(f'(x) = 0\):
- The derivative \(f'(x)\) is undefined at \(x = 2\).
Hence, the critical points are:
\[ x = 1 \quad \text{and} \quad x = 2. \]
Final Answer: 2.
Top JEE Main Mathematics Questions
- The value of is
- The number of terms of an is even. The sum of the odd terms is and of the even terms is . The last term exceeds the first by . Then the number of terms in the series is ______.
- If , then the general value of is:
- The general solution of
- If the constant term in the binomial expansion of $\left(\frac{x^{\frac{3}{2}}}{2}-\frac{4}{x^l}\right)^9$ is $-84$ and the coefficient of $x^{-3 l}$ is $2^\alpha \beta$, where $\beta<0$ is an odd number, then $|\alpha l-\beta|$ is equal to_____
Top JEE Main Maxima and Minima Questions
- Let the set of all positive values of \( \lambda \), for which the point of local minimum of the function \((1 + x (\lambda^2 - x^2)) \frac{x^2 + x + 2}{x^2 + 5x + 6} < 0\) be \((\alpha, \beta)\).
Then \( \alpha^2 + \beta^2 \) is equal to ________. - If the function \(f(x) = 2x^3 - 9ax^2 + 12a^2x + 1, \, a>0\) has a local maximum at \(x = \alpha\) and a local minimum at \(x = \alpha^2\), then \(\alpha\) and \(\alpha^2\) are the roots of the equation:
- Let $f(x) = 4\cos^3 x + 3\sqrt{3} \cos^2 x - 10$. The number of points of local maxima of $f$ in interval $(0, 2\pi)$ is:
- Let \( f(x) = 3\sqrt{x - 2} + \sqrt{4 - x} \) be a real-valued function. If \( \alpha \) and \( \beta \) are respectively the minimum and the maximum values of \( f \), then \( \alpha^2 + 2\beta^2 \) is equal to:
- Let the maximum and minimum values of \[\left( \sqrt{8x - x^2 - 12 - 4} \right)^2 + (x - 7)^2, \quad x \in \mathbb{R} \text{ be } M \text{ and } m \text{ respectively}.\] Then \( M^2 - m^2 \) is equal to _____.
Top JEE Main Questions
- Two masses $m$ and $\frac{m}{2}$ are connected at the two ends of a massless rigid rod of length $l$. The rod is suspended by a thin wire of torsional constant $k$ at the centre of mass of the rod-mass system(see figure). Because of torsional constant $k$, the restoring torque is $\tau =k\theta$ for angular displacement $\theta$. If the rod is rota ted by $\theta_0$ and released, the tension in it when it passes through its mean position will be:
What will be the equilibrium constant of the given reaction carried out in a \(5 \,L\) vessel and having equilibrium amounts of \(A_2\) and \(A\) as \(0.5\) mole and \(2 \times 10^{-6}\) mole respectively?
The reaction : \(A_2 \rightleftharpoons 2A\)- The value of is
- The number of terms of an is even. The sum of the odd terms is and of the even terms is . The last term exceeds the first by . Then the number of terms in the series is ______.
A black body is at a temperature of 2880 K. The energy of radiation emitted by this body with wavelength between 499 nm and 500 nm is U1, between 999 nm and 1000 nm is U2 and between 1499 nm and 1500 nm is U3. The Wien's constant, b = 2.88×106 nm-K. Then,