Question Type 2: Checking whether the first order conditions constitute to a local minimum or maximum Exercises

Consider $f(x)=x^3-3x$. The first-order condition $f'(x)=0$ gives $x=\pm 1$. Classify each stationary point as a local minimum, local maximum, or neither.

For $f(x)=x^4$, the first-order condition $f'(x)=0$ gives $x=0$. Determine whether $x=0$ is a local minimum, local maximum, or neither.

Let $f(x)=x e^{-x}$ on $x\ge 0$. The first-order condition $f'(x)=0$ gives $x=1$. Determine whether this stationary point is a local minimum, local maximum, or neither.

Find the $x$-coordinates of the critical points of the function $f(x)=2x^3-6x^2+3$, and determine whether each is a local minimum or maximum.

On the domain $x>0$, let $f(x)=x+\frac{1}{x}$. The first-order condition $f'(x)=0$ gives $x=1$. Determine whether this stationary point is a local minimum, local maximum, or neither.

Determine whether the function $f(x)=x^5-5x^3+4x$ has a local extremum at $x=0$.

Consider $f(x)=x^4-4x^2+1$. Find all critical points and classify them as local minima or maxima.

On the domain $x>0$, consider $f(x)=\ln x - x^2$. The first-order condition $f'(x)=0$ gives $x=\frac{1}{\sqrt{2}}$. Determine whether this stationary point is a local minimum, local maximum, or neither.

For $f(x)=\sin x+\cos x$ on $[0,2\pi]$, the first-order condition $f'(x)=0$ yields $x=\frac{\pi}{4},\,\frac{5\pi}{4}$. Classify each as a local minimum, local maximum, or neither.

Let $f(x)=x^3-3ax$ with parameter $a\in\mathbb{R}$. The first-order condition $f'(x)=0$ gives critical points depending on $a$. Classify the stationary points as local minima, local maxima, or neither in terms of $a$.

Given $f(x) = x^2 - 4x + 5$, the first-order condition $f'(x)=0$ yields $x=2$. Determine whether this stationary point is a local minimum, local maximum, or neither.

For $f(x)=x^3$, the first-order condition $f'(x)=0$ gives $x=0$. Decide whether this stationary point is a local minimum, local maximum, or neither.