Mean Value Theorem and Extreme Value Theorem: Practice Questions

Consider the function \( f(x) = x^4 - 4x^3 + 10 \). On which open interval is the graph of \( f \) concave upward?

The function \( f \) is given by \( f(x) = x^4 - 6x^2 + 8x + 10 \). On which of the following intervals is the graph of \( f \) concave downward?

A rectangle is inscribed under the curve \(y = e^{-x^2}\) with its base on the \(x\)-axis and symmetric about the \(y\)-axis. What is the maximum possible area of this rectangle?

Consider the function \( f(x) = x^3 + 3x^2 - 9x + 5 \). On which of the following open intervals is the function \( f \) strictly decreasing?

The graph of the second derivative, \( f''(x) \), of a function is given as \( f''(x) = (x+1)(x-2)^2 \). At which value(s) of \( x \) does the graph of \( f(x) \) have a point of inflection?

Consider the function \( f(x) = x^4 - 2x^2 + 3 \). Find the \( x \)-coordinates of all critical points of \( f(x) \).

A particle moves along the \( x \)-axis with a velocity given by \( v(t) = t^2 - 7t + 10 \) for \( t \ge 0 \). At what time \( t \) does the particle reach its minimum velocity?

Let \( f(x) = x^3 + ax^2 + bx \). If the graph of \( f \) has a relative maximum at \( x = -1 \) and a point of inflection at \( x = 1 \), find the values of the constants \( a \) and \( b \).

Consider the function \( f(x) = x^3 - 6x^2 + 9x + 2 \).
a) Find the intervals on which \( f \) is increasing and the intervals on which \( f \) is decreasing.
b) Use the results from part (a) to find the \( x \)-coordinates of all relative extrema.

Let \( f(x) = x \ln(x) \) for \( x > 0 \).
a) Determine the absolute minimum value of \( f \) on its domain. Justify your answer.
b) Show that the graph of \( f \) is concave upward for all \( x \) in its domain.
c) Use the results to determine the number of solutions to the equation \( x \ln(x) = 2 \).