Problem 77 Suppose $$ p(x)=x^{5}+2 x^{3... [FREE SOLUTION]

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Precalculus A Prelude to Calculus

Sheldon Axler

$Math Studyset 91影视 Explanations$ Math

2 Edition

Chapter 2: Problem 77

Suppose $$ p(x)=x^{5}+2 x^{3}+1 $$ (a) Find two distinct points on the graph of $p$. (b) Explain why $p$ is an increasing function. (c) Find two distinct points on the graph of $p^{-1}$.

Short Answer

Expert verified

We found two points on the graph of $p(x)$: $(0, 1)$ and $(1, 4)$. The function $p(x)$ is an increasing function because its derivative, $p'(x) = 5x^4 + 6x^2$, is always non-negative. Two points on the graph of $p^{-1}(x)$ are $(1, 0)$ and $(4, 1)$.

Step by step solution

Find two points on the graph of $p(x)$

To find two points on the graph of this function, we can evaluate the function for any two distinct $x$ values. For simplicity, let's consider $x=0$ and $x=1$. For $x = 0$, we have: $p(0) = (0)^{5} + 2(0)^{3} + 1 =0 + 0 + 1 = 1$ Thus, the point $(0,1)$ is on the graph of $p(x)$. For $x = 1$, we have: $p(1) = (1)^{5} + 2(1)^{3} + 1 = 1 + 2 + 1 = 4$ Thus, the point $(1,4)$ is on the graph of $p(x)$.

Explain why $p$ is an increasing function

In order to determine if the function $p(x) = x^5 + 2x^3 + 1$ is an increasing function, we can examine its derivative. Recall that if the derivative is positive, it means the function is increasing. So, let's differentiate $p(x)$ with respect to $x$. $ p'(x) = \frac{d}{dx}(x^5 + 2x^3 + 1) = 5x^4 + 6x^2 $ Now observe that both terms in p'(x) are non-negative since $5x^4$ and $6x^2$ are both positive or zero for any real value of x. Hence, the derivative $p'(x)$ is always non-negative, which implies the function $p(x)$ is increasing.

Find two points on the graph of $p^{-1}(x)$

To find two points on $p^{-1}(x)$, we can use the fact that if $(a,b)$ is on the graph of a function $p(x)$, then $(b,a)$ is on the graph of its inverse function $p^{-1}(x)$. So, we already have two points on the graph of $p(x)$: $(0,1)$ and $(1,4)$. Therefore, the points $(1,0)$ and $(4,1)$ must be on the graph of $p^{-1}(x)$.

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91影视

Suppose $$ p(x)=x^{5}+2 x^{3}+1 $$ (a) Find two distinct points on the graph of \(p\). (b) Explain why \(p\) is an increasing function. (c) Find two distinct points on the graph of \(p^{-1}\).

Short Answer

Step by step solution

Find two points on the graph of \(p(x)\)

Explain why \(p\) is an increasing function

Find two points on the graph of \(p^{-1}(x)\)

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