Question

attempt 1: 10 attempts remaining. let $f(x)=x^{4}+5x^{3}+2x^{2}+2x$. $f(x)=$ $f(2)=$ $f(x)=$ $f(2)=$ submit answer next item

Explanation:

Step1: Apply power - rule for differentiation

The power - rule states that if $y = x^n$, then $y^\prime=nx^{n - 1}$. For $f(x)=x^{4}+5x^{3}+2x^{2}+2x$, we have:
$f^\prime(x)=\frac{d}{dx}(x^{4})+\frac{d}{dx}(5x^{3})+\frac{d}{dx}(2x^{2})+\frac{d}{dx}(2x)$.
Using the constant - multiple rule $\frac{d}{dx}(cf(x)) = c\frac{d}{dx}(f(x))$ and power - rule, we get $f^\prime(x)=4x^{3}+15x^{2}+4x + 2$.

Step2: Evaluate $f^\prime(2)$

Substitute $x = 2$ into $f^\prime(x)$:
$f^\prime(2)=4\times2^{3}+15\times2^{2}+4\times2 + 2=4\times8+15\times4 + 8+2=32 + 60+8 + 2=102$.

Step3: Differentiate $f^\prime(x)$ to get $f^{\prime\prime}(x)$

Differentiate $f^\prime(x)=4x^{3}+15x^{2}+4x + 2$ using the power - rule.
$f^{\prime\prime}(x)=\frac{d}{dx}(4x^{3})+\frac{d}{dx}(15x^{2})+\frac{d}{dx}(4x)+\frac{d}{dx}(2)=12x^{2}+30x + 4$.

Step4: Evaluate $f^{\prime\prime}(2)$

Substitute $x = 2$ into $f^{\prime\prime}(x)$:
$f^{\prime\prime}(2)=12\times2^{2}+30\times2 + 4=12\times4+60 + 4=48+60 + 4=112$.

Answer:

$f^\prime(x)=4x^{3}+15x^{2}+4x + 2$
$f^\prime(2)=102$
$f^{\prime\prime}(x)=12x^{2}+30x + 4$
$f^{\prime\prime}(2)=112$