5. Graphical Applications of Derivatives

For the function $f\left(x\right)$ defined on the interval $[0,6]$, the graph shows a peak at $x=2$ and a trough at $x=5$. Which interval contains the local maximum of $f\left(x\right)$?

130. Where does the periodic function f(x) = 2e^(sin(x/2)) take on its extreme values, and what are these values?

Find the critical points of the following functions on the given intervals. Identify the absolute maximum and absolute minimum values (if they exist).

ƒ(x) = sin 2x + 3 on [-π , π]

Finding Extrema from Graphs

In Exercises 7–10, find the absolute extreme values and where they occur.

Find the critical points of the following functions on the given intervals. Identify the absolute maximum and absolute minimum values (if they exist).

ƒ(x) = 4x¹⸍² - x⁵⸍² on [0, 4]

Find the critical points of the following functions on the given intervals. Identify the absolute maximum and absolute minimum values (if they exist).

ƒ(x) = x³ - 6x² on [-1, 5]

Find the global maximum and minimum values of the function $f(x,y)=e^{-xy}$ on the region defined by $x^2+4y^2\le 1$.

Let $f\left(x\right)=x^3-3x^2+1$. What is the absolute minimum value of $f$ on the closed interval $[0,3]$?

What is the end behavior of the graph of the polynomial function $y=10x^9-4x$?

For the curve $y=1+40x^3-3x^5$, at which point does the tangent line have the largest slope?

Absolute Extrema on Finite Closed Intervals

In Exercises 21–36, find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates.

f(x) = (2/3)x − 5, −2 ≤ x ≤ 3

Let $f\left(x\right)=x^3+a{x}^2$. For what value of $a$ does $f\left(x\right)$ have a critical point at $x=1$?

Let $f\left(x\right)$ = $x^3-3x^2+4$ on the interval $[0,3]$. What is the maximum value of $f$ on this interval?

For the curve $y=5\ln \left(x\right)$, at what point does the curve have maximum curvature?

Over which interval is the graph of $f\left(x\right)=\frac{1}{2}{x}^2+5x+6$ increasing?