Question

7 mark for review
what is the x-coordinate of the point of inflection on the graph of $y = \frac{1}{3}x^3 + 5x^2 + 24$?
a 5
b 0
c $-\frac{10}{3}$
d -5
e -10

Explanation:

Step1: Recall Inflection Point Rule

A point of inflection occurs where the second derivative changes sign, so we first find the second derivative of \( y=\frac{1}{3}x^{3}+5x^{2}+24 \).

Step2: Find First Derivative

Using the power rule \( \frac{d}{dx}(x^n)=nx^{n - 1} \), the first derivative \( y' \) is:
\( y'=\frac{d}{dx}(\frac{1}{3}x^{3}+5x^{2}+24)=3\times\frac{1}{3}x^{2}+2\times5x+0=x^{2}+10x \)

Step3: Find Second Derivative

Differentiate \( y' = x^{2}+10x \) again:
\( y''=\frac{d}{dx}(x^{2}+10x)=2x + 10 \)

Step4: Solve \( y'' = 0 \) for \( x \)

Set \( 2x+10 = 0 \). Subtract 10 from both sides: \( 2x=-10 \). Divide by 2: \( x=-5 \). We check that the second derivative changes sign at \( x = - 5 \) (for \( x<-5 \), say \( x=-6 \), \( y''=2(-6)+10=-2<0 \); for \( x>-5 \), say \( x = 0 \), \( y''=10>0 \)), so it's a point of inflection.

Answer:

D. -5