The number of critical points of the function $f(x)=(x-2)^{2 / 3}(2 x+1)$ is

<p>To find the number of critical points of the function $f(x)=(x-2)^{2 / 3}(2 x+1)$, we need to determine where its derivative $f'(x)$ is equal to zero or undefined. Critical points occur where the derivative is zero or does not exist.</p> <p>First, let's find the derivative of the function:</p> <p>$f(x)=(x-2)^{2 / 3}(2 x+1)$</p> <p>We apply the product rule for differentiation, which states that $(uv)' = u'v + uv'$, where $u = (x-2)^{2/3}$ and $v = 2x + 1$.</p> <p>We need the derivatives of $u$ and $v$:</p> <p>For $u = (x-2)^{2/3}$, we use the chain rule:</p> <p>$$u'(x) = \frac{d}{dx}[(x-2)^{2/3}] = \frac{2}{3}(x-2)^{-1/3} \cdot 1 = \frac{2}{3}(x-2)^{-1/3}$$</p> <p>The derivative of $v$ is straightforward, as $v = 2x + 1$:</p> <p>$v'(x) = 2$</p> <p>Now we apply the product rule:</p> <p>$f'(x) = u'(x)v(x) + u(x)v'(x)$</p> <p>Substituting $u$, $u'$, and $v$, we get:</p> <p>$$f'(x) = \left( \frac{2}{3}(x-2)^{-1/3} \right)(2x+1) + \left( (x-2)^{2/3} \right)(2)$$</p> <p>This simplifies to:</p> <p>$f'(x) = \frac{2(2x + 1)}{3(x-2)^{1/3}} + 2(x-2)^{2/3}$</p> <p>For critical points, we need to solve $f'(x) = 0$ or where it is undefined.</p> <p>1. <strong>Solve for where the derivative is zero:</strong></p> <p>$\frac{2(2x + 1)}{3(x-2)^{1/3}} + 2(x-2)^{2/3} = 0$</p> <p>Combining like terms in a common denominator, we get:</p> <p>$\frac{2(2x + 1) + 6(x - 2)}{3(x-2)^{1/3}} = 0$</p> <p>Simplifying the numerator:</p> <p>$2(2x + 1) + 6(x-2) = 4x + 2 + 6x - 12 = 10x - 10 = 10(x-1)$</p> <p>So:</p> <p>$\frac{10(x-1)}{3(x-2)^{1/3}} = 0$</p> <p>The numerator is zero when:</p> <p>$10(x-1) = 0$</p> <p>Therefore:</p> <p>$x = 1$</p> <p>2. <strong>Solve for where the derivative is undefined:</strong></p> <p>The denominator, $3(x-2)^{1/3}$, is undefined when $(x-2)^{1/3} = 0$, which happens at:</p> <p>$x = 2$</p> <p>From the above analysis, the critical points are at $x = 1$ and $x = 2$. Thus, there are 2 critical points.</p> <p>Therefore, the number of critical points of the function $f(x)=(x-2)^{2 / 3}(2 x+1)$ is:</p> <p><b>Option A</b></p>