use graphing technology (e.g., desmos) to help answer the following question(s). round your answer(s) to…

use graphing technology (e.g., desmos) to help answer the following question(s). round your answer(s) to three decimal places.\n\n$f(x)=\\sqrt{7 x^{3}+11 x^{2}}$\n\n a. the function has a local maximum of $y=\\square$ at $x=\\square$\n b. the function has a local minimum of $y=\\square$ at $x=\\square$\n c. the function is increasing on the interval(s) $\\square$\n d. the function is decreasing on the interval(s) $\\square$\n\n question help: video written example

use graphing technology (e.g., desmos) to help answer the following question(s). round your answer(s) to three decimal places.\n\n$f(x)=\\sqrt{7 x^{3}+11 x^{2}}$\n\n a. the function has a local maximum of $y=\\square$ at $x=\\square$\n b. the function has a local minimum of $y=\\square$ at $x=\\square$\n c. the function is increasing on the interval(s) $\\square$\n d. the function is decreasing on the interval(s) $\\square$\n\n question help: video written example

Answer

Explanation:

Step1: Analyze the function ( f(x)=\sqrt{7x^{3}+11x^{2}} )

First, find the domain. For the square - root function ( y = \sqrt{u} ), ( u=7x^{3}+11x^{2}=x^{2}(7x + 11)\geq0 ). The solutions of ( x^{2}(7x + 11)\geq0 ) are ( x\geq-\frac{11}{7} ).

Step2: Use a graphing utility (e.g., Desmos)

Input the function ( y=\sqrt{7x^{3}+11x^{2}} ) into the graphing utility.

Part a

By looking at the graph, we find the local maximum. The local maximum occurs at ( x =-\frac{22}{21}\approx - 1.048 ). Substitute ( x =-\frac{22}{21} ) into ( f(x) ): ( f\left(-\frac{22}{21}\right)=\sqrt{7\left(-\frac{22}{21}\right)^{3}+11\left(-\frac{22}{21}\right)^{2}}=\sqrt{\frac{-7\times22^{3}+11\times22^{2}\times21}{21^{3}}}=\sqrt{\frac{22^{2}(-7\times22 + 11\times21)}{21^{3}}}=\sqrt{\frac{22^{2}( - 154+231)}{21^{3}}}=\sqrt{\frac{22^{2}\times77}{21^{3}}}\approx1.591 )

Part b

The local minimum occurs at ( x = 0 ). Substitute ( x = 0 ) into ( f(x) ), ( f(0)=\sqrt{7\times0^{3}+11\times0^{2}} = 0 )

Part c

To find where the function is increasing, we look at the intervals on the graph where the function moves up as ( x ) increases. The function is increasing on the interval ( \left(-\frac{11}{7},-\frac{22}{21}\right)\cup(0,\infty) )

Part d

To find where the function is decreasing, we look at the intervals on the graph where the function moves down as ( x ) increases. The function is decreasing on the interval ( \left(-\frac{22}{21},0\right) )

Answer:

a. ( y\approx1.591 ) at ( x\approx - 1.048 ) b. ( y = 0 ) at ( x = 0 ) c. The function is increasing on the interval(s) ( \left(-\frac{11}{7},-\frac{22}{21}\right)\cup(0,\infty) ) d. The function is decreasing on the interval(s) ( \left(-\frac{22}{21},0\right) )