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ololo11 [35]
2 years ago
10

Help please!figure out the "inflection point" of the following functionf(x)=5x³ + 2x² − 3x​

Mathematics
2 answers:
Artyom0805 [142]2 years ago
3 0

Answer:

x = -2/15.

Step-by-step explanation:

Finding the second derivative:

f'(x) = 15x^2 +4x - 3

f"(x) = 30x + 4

30x + 4 = 0   for a point of inflection

x = -4/30 =  -2/15

So 30x + 4 is negative up to x =  -2/15 and positive thereafter.

So the curve passes from concave downward to concave upward at x = -2/15.

aivan3 [116]2 years ago
3 0

Answer:

\displaystyle \left(  - \frac{2}{15} , \frac{286}{675}  \right)

Step-by-step explanation:

to figure out the <u>infection</u><u> </u><u>point </u>

take derivative both sides:

\displaystyle f'(x) =  \frac{d}{dx}  {5x}^{3}  + 2 {x}^{2}  - 3x

By sum derivation rule we acquire:

\displaystyle  \rm f'(x) =  \frac{d}{dx}  {5x}^{3}  + \frac{d}{dx}  2 {x}^{2}  - \frac{d}{dx}  3x

apply exponent derivation rule which yields:

\displaystyle   f'(x) =    {15x}^{2}  + 4{x}^{}  -   3

take derivative in both sides once again which yields:

\rm\displaystyle   f''(x) =     \frac{d}{dx} {15x}^{2}  +  \frac{d}{dx} 4{x}^{}  -    \frac{d}{dx} 3

remember that, derivative of a constant is always 0 so,

\rm\displaystyle   f''(x) =     \frac{d}{dx} {15x}^{2}  +  \frac{d}{dx} 4{x}^{}  -    0

by exponent derivation rule we acquire:

\rm\displaystyle   f''(x) =     {30x}  +  4{}^{}

substitute f''(x) to 0 figure out the x coordinate of the inflection point:

\rm\displaystyle      {30x}  +  4{}^{}  = 0

cancel 4 from both sides:

\rm\displaystyle      {30x}  =  - 4

divide both sides by 30:

\rm\displaystyle      {x}  =     - \frac{2}{15}

now plugin the value of x to the given function to figure out the y coordinate of the inflection point:

\rm \displaystyle f(x) =  {5 \left( -  \frac{2}{15} \right) }^{3}  + 2 {\left(  - \frac{2}{15} \right) }^{2}   - 3 \left(  - \frac{2}{15}  \right)

By simplifying we acquire:

\displaystyle f(x) =  \frac{286}{675}

hence,

the coordinates of inflection point are

\displaystyle \left(  - \frac{2}{15} , \frac{286}{675}  \right)

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