Answer :
In order to get the maximum (or minimum) of a curve, we need to get the first derivative of the function or the slope then equate to zero. In order to confirm if the value of x is really the maxima, we can proceed to the second derivative and substitute the x to the equation. If the result is a negative number, then the point is the maxima. Therefore,
f(x) = (x^4/3) - (x^5/5)
f'(x) = (4x^3/3) - x^4 = 0
thus x = 4/3
check if the point is really the maxima
f"(x) = 4x^2 - 4x^3
substituting x = 4/3, we get an answer of -2.3704 thus the point is a maxima.
f(x) = (x^4/3) - (x^5/5)
f'(x) = (4x^3/3) - x^4 = 0
thus x = 4/3
check if the point is really the maxima
f"(x) = 4x^2 - 4x^3
substituting x = 4/3, we get an answer of -2.3704 thus the point is a maxima.
The value of x is [tex]\boxed{\frac{4}{3}}[/tex] for which the derivative of [tex]f\left( x \right) =\dfrac{{{x^4}}}{3} - \dfrac{{{x^5}}}{5}[/tex] attains the maximum.
Further explanation:
Given:
The function is [tex]f\left( x \right) =\dfrac{{{x^4}}}{3} - \dfrac{{{x^5}}}{5}.[/tex]
Explanation:
The given function is [tex]f\left( x \right)=\dfrac{{{x^4}}}{3} - \dfrac{{{x^5}}}{5}.[/tex]
Differentiate the above equation with respect to x.
[tex]\begin{aligned}\frac{d}{{dx}}f\left( x \right) &= \frac{d}{{dx}}\left( {\frac{{{x^4}}}{3} - \frac{{{x^5}}}{5}} \right)\\&= \frac{{4{x^3}}}{3} - \frac{{5{x^4}}}{5}\\&= \frac{{4{x^3}}}{3} - {x^4}\\\end{aligned}[/tex]
Again differentiate with respect to x.
[tex]\begin{aligned}\frac{{{d^2}}}{{d{x^2}}}f\left( x \right) &= \frac{{{d^2}}}{{d{x^2}}}\left( {\frac{{4{x^3}}}{3} - {x^4}} \right)\\&=\frac{{3 \times 4{x^2}}}{3} - 4{x^3}\\&= 4{x^2} - 4{x^3}\\\end{aligned}[/tex]
Substitute the first derivative equal to zero.
[tex]\begin{aligned}\frac{d}{{dx}}f\left( x \right)&= 0\\\frac{{4{x^3}}}{3} - {x^4}&= 0\\\frac{{4{x^3}}}{3} &= {x^4}\\\frac{4}{3}&= \frac{{{x^4}}}{{{x^3}}}\\\frac{4}{3}&= x\\\end{aligned}[/tex]
The value of x is [tex]\boxed{\frac{4}{3}}[/tex] for which the derivative of [tex]f\left( x \right)=\dfrac{{{x^4}}}{3} - \dfrac{{{x^5}}}{5}[/tex] attains the maximum.
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Answer details:
Grade: High School
Subject: Mathematics
Chapter: Application of derivatives
Keywords: Derivative, attains, maximum, value of x, function, differentiate, minimum value.