Question

Local Maximum and Minimum Values The graph of a function f is given. Use the graph to estimate the following. (a) All the local maximum and minimum values of the function and the value of x at which each occurs. (b) The intervals on which the function is increasing and on which the function is decreasing.

Short answer

(a) The local maximum value $0$is $1$and $x=0$at and$x=3$

The local minimum value is $-2$at $x=-2$.

(b) The function f is increasing on $(-2,0)\cup (1,3)$.

The function f is decreasing on $(-\infty ,-2)\cup (0,1)\cup (3,\infty )$.

Step-by-step solution

Part a. Step 1. Concept of local maximum.

The function value $f\left(a\right)$is a local maximum value of$f$ if $f\left(a\right)\ge f\left(x\right)$when$x$ is a near $a$.

Part a. Step 2. Concept of local maximum.

The function value $f\left(a\right)$is a local minimum value of if $f\left(a\right)\le f\left(x\right)$ when$x$ is a near $a$.

Part a. Step 3. Determining local maximum and local minimum.

The local maximum value is$0$at$x=0$and the local maximum value is$1$at$x=3$.

The local minimum value is $-2$at $x=-2$.

Part b. Step 1. Concept of increasing function.

The function$f$ is said to be increasing when the value of$y$ continuously increases that is $f\left(y_1\right)<f\left(y_2\right)$.

Where $y_1$and$y_2$ is at $y$–axis such that$y_1<y_2.$

Part b. Step 2. Concept of decreasing function.

The function$f$ is said to be decreasing when the value of$y$ continuously decreases that is $f\left(y_1\right)>f\left(y_2\right)$.

Where $y_1$and$y_2$ is at $y$–axis such that$y_1<y_2.$

Part b. Step 3. Determining the interval on which f is increasing and decreasing.

The function f is increasing on$(-2,0)\cup (1,3)$.

The function f is decreasing on $(-\infty ,-2)\cup (0,1)\cup (3,\infty )$.