Question

find ( f(x) ).

( f(x)=4x^{15}-6x^{3}+5 )

( f(x)=)

compare the graphs of ( f ) and ( f ) and use them to explain why your answer is reasonable.

( f(x) = 0 ) when ( f ) ---select---.

( f ) is positive when ( f ) ---select---.

( f ) is negative when ( f ) ---select---.

Explanation:

Step1: Apply power - rule for differentiation

The power - rule states that if $y = ax^n$, then $y^\prime=anx^{n - 1}$, and the derivative of a constant $C$ is $0$. For $f(x)=4x^{15}-6x^{3}+5$, the derivative of $4x^{15}$ is $4\times15x^{15 - 1}=60x^{14}$, the derivative of $-6x^{3}$ is $-6\times3x^{3 - 1}=-18x^{2}$, and the derivative of the constant $5$ is $0$.

Step2: Combine the derivatives

$f^\prime(x)=60x^{14}-18x^{2}$

For the second part:

• $f^\prime(x) = 0$ when $f$ has a horizontal tangent (local maximum, local minimum, or an inflection point with a horizontal tangent).
• $f^\prime$ is positive when $f$ is increasing.
• $f^\prime$ is negative when $f$ is decreasing.

Answer:

$f^\prime(x)=60x^{14}-18x^{2}$
$f^\prime(x) = 0$ when $f$ has a horizontal tangent.
$f^\prime$ is positive when $f$ is increasing.
$f^\prime$ is negative when $f$ is decreasing.