Question Type 1: Finding the first order conditions for polynomials up to cubics Exercises

Find the critical points of $f(x) = x^3 e^{-x^2}$.

Find the stationary point of $f(x) = x^2 - 8x + 15$.

Find the stationary points of $f(x) = \frac{x^3}{3} + \frac{5x^2}{2} + 6x + 10$.

For $f(x) = x^3 - 3x^2 + 2$, determine whether each stationary point is a local minimum or maximum.

Find all critical points of $f(x)=3x^3-6x^2+9x-2$.

Find the critical points for the function $f(x) = x^3 e^{-x^2}$ and classify each as a local maximum, a local minimum, or a horizontal point of inflection.

Determine the stationary points of $f(x) = -x^3 + 4x^2 - x + 1$.

Find the critical points of $f(x) = \frac{x^2 e^x}{x - 1}$ using technology, and give real solutions accurate to two decimal places.

The function $f(x) = e^x + x - \ln(x)$ has a single critical point at $x \approx 0.402$. Determine whether this critical point is a local minimum or a local maximum.

Find the critical point(s) of $f(x) = e^x + x - \ln(x)$ using technology and give your answer to three decimal places.

For the function $f(x) = \frac{x^2e^x}{x^2+1}$, determine whether each critical point is a local minimum or a local maximum. Justify your answer using a sign chart or technology.

Find the critical points of the function $f(x) = \frac{x^2 e^x}{x^2+1}$. Give your answers as coordinates $(x, y)$, approximating all values to two decimal places where necessary.