consider the following function. use a graphing utility to confirm your answers for parts (a) through (c)…

consider the following function. use a graphing utility to confirm your answers for parts (a) through (c). (if an answer does not exist, enter dne.)\n\n$f(x)=x^{1/2}+8$\n\n(a) find the critical numbers of f. (enter your answers as a comma-separated list.)\n\n$x=$\n\n(b) find the open intervals on which the function is increasing or decreasing. (enter your answers using interval notation.)\n\nincreasing\n\ndecreasing\n\n(c) apply the first derivative test to identify all relative extrema.\n\nrelative maximum $(x,y)=()$\n\nrelative minimum $(x,y)=()$

consider the following function. use a graphing utility to confirm your answers for parts (a) through (c). (if an answer does not exist, enter dne.)\n\n$f(x)=x^{1/2}+8$\n\n(a) find the critical numbers of f. (enter your answers as a comma-separated list.)\n\n$x=$\n\n(b) find the open intervals on which the function is increasing or decreasing. (enter your answers using interval notation.)\n\nincreasing\n\ndecreasing\n\n(c) apply the first derivative test to identify all relative extrema.\n\nrelative maximum $(x,y)=()$\n\nrelative minimum $(x,y)=()$

Answer

Explanation:

Step1: Find the derivative

Using the power rule (y = x^n), (y^\prime=nx^{n - 1}), for (f(x)=x^{\frac{1}{2}}+8), (f^\prime(x)=\frac{1}{2}x^{-\frac{1}{2}}=\frac{1}{2\sqrt{x}})

Step2: Find critical numbers

Critical numbers occur where (f^\prime(x) = 0) or (f^\prime(x)) is undefined. Set (f^\prime(x)=0), (\frac{1}{2\sqrt{x}} = 0) has no solution. (f^\prime(x)) is undefined when (x = 0) (since (\sqrt{x}) is undefined for (x<0) in the real - valued function and at (x = 0), the derivative has a non - removable discontinuity). So the critical number is (x = 0)

Step3: Determine intervals of increase/decrease

Choose a test point in the interval ((0,\infty)). Let (x = 1), then (f^\prime(1)=\frac{1}{2\sqrt{1}}=\frac{1}{2}>0) The function (f(x)) is defined for (x\geq0). The domain of (f(x)) is ([0,\infty)). Since (f^\prime(x)>0) for (x\in(0,\infty)), the function is increasing on ((0,\infty)) and there is no interval where (f(x)) is decreasing (because the domain starts at (x = 0) and the derivative is positive for (x>0))

Step4: Apply the First Derivative Test for extrema

Since the function changes from not being differentiable (at the left - hand "edge" of its domain) to increasing (for (x>0)), and (f(0)=0^{\frac{1}{2}}+8=8) There is no relative maximum (because the function only starts at (x = 0) and then increases). The relative minimum occurs at ((x,y)=(0,8))

Answer:

(a) (x = 0) (b) increasing: ((0,\infty)), decreasing: (\text{NONE}) (c) relative maximum: (\text{NONE}), relative minimum: ((0,8))