let f(x)=-2x^6 + 15x^5. for what values of x does the graph of f have a point of inflection? choose all…

let f(x)=-2x^6 + 15x^5. for what values of x does the graph of f have a point of inflection? choose all answers that apply: a x = 0 b x = 5 c x = 15/2 d f has no points of inflection.

let f(x)=-2x^6 + 15x^5. for what values of x does the graph of f have a point of inflection? choose all answers that apply: a x = 0 b x = 5 c x = 15/2 d f has no points of inflection.

Answer

Explanation:

Step1: Find the first - derivative

Using the power rule $\frac{d}{dx}(x^n)=nx^{n - 1}$, for $f(x)=-2x^{6}+15x^{5}$, we have $f'(x)=-12x^{5}+75x^{4}$.

Step2: Find the second - derivative

Differentiate $f'(x)$ again using the power rule. $f''(x)=-60x^{4}+300x^{3}$.

Step3: Set the second - derivative equal to zero

Set $f''(x) = 0$, so $-60x^{4}+300x^{3}=0$. Factor out $- 60x^{3}$: $-60x^{3}(x - 5)=0$.

Step4: Solve for x

Set each factor equal to zero. For $-60x^{3}=0$, we get $x = 0$. For $x - 5=0$, we get $x = 5$.

Answer:

A. $x = 0$, B. $x = 5$