Answered

IDNLearn.com: Your trusted source for accurate and reliable answers. Get thorough and trustworthy answers to your queries from our extensive network of knowledgeable professionals.

The polynomial P(x) = 5x^3 + 2x^2- 42 has ___ local maxima and minima

The Polynomial Px 5x3 2x2 42 Has Local Maxima And Minima class=

Sagot :

SOLUTION:

Case: Local maxima and local minima

Given:

[tex]P(x)=5x^3+2x^2-42^{}[/tex]

Required: To find the local maxima and minima

Method:

[tex]\begin{gathered} P(x)=5x^3+2x^2-42^{} \\ \text{ } \end{gathered}[/tex]

First the find the first derivative:

[tex]\begin{gathered} P^{\prime}(x)=15x^2+4x \\ we\text{ make P'(x) = 0} \\ 15x^2+4x=\text{ 0} \\ x(15x+4)=0_{} \\ x=0\text{ or 15x+4 =0} \\ x=\text{ 0 or x= }\frac{-4}{15} \end{gathered}[/tex]

Let us take the points in the immediate neighbourhood of x = 0. The points are {-1, 1}.

[tex]\begin{gathered} \text{when x=-1} \\ P^{\prime}(x)=15x^2+4x \\ P^{\prime}(-1)=15(-1)^2+4(-1) \\ P^{\prime}(-1)=15(1)^2-4 \\ P^{\prime}(-1)=15^{}-4 \\ P^{\prime}(-1)=11 \\ \text{when x= 1} \\ P^{\prime}(x)=15x^2+4x \\ P^{\prime}(1)=15(1)^2+4(1) \\ P^{\prime}(1)=15^{}+4 \\ P^{\prime}(1)=19 \end{gathered}[/tex]

Let us take the points in the immediate neighbourhood of x = -4/15. The points are {-1, 0}

[tex]\begin{gathered} \text{when x= 0} \\ P^{\prime}(x)=15x^2+4x \\ P^{\prime}(0)=15(0)^2+4(0) \\ P^{\prime}(0)=0^{}+0 \\ P^{\prime}(0)=0^{} \end{gathered}[/tex]

Final answer:

The local minima is at x= 0 while the local maxima is at x= -4/15

Thank you for using this platform to share and learn. Keep asking and answering. We appreciate every contribution you make. Find clear and concise answers at IDNLearn.com. Thanks for stopping by, and come back for more dependable solutions.