Question

suppose that $f(x)=3x^{4/5}-4x^{5/7}$. evaluate each of the following: $f(2)=$ $f(5)=$

Explanation:

Step1: Find the derivative of \(f(x)\)

Use the power - rule \((x^n)^\prime=nx^{n - 1}\).
For \(y = 3x^{4/5}-4x^{5/7}\), \(y^\prime=f^\prime(x)=3\times\frac{4}{5}x^{\frac{4}{5}-1}-4\times\frac{5}{7}x^{\frac{5}{7}-1}\)
\(f^\prime(x)=\frac{12}{5}x^{-\frac{1}{5}}-\frac{20}{7}x^{-\frac{2}{7}}\)

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

Substitute \(x = 2\) into \(f^\prime(x)\).
\(f^\prime(2)=\frac{12}{5}\times2^{-\frac{1}{5}}-\frac{20}{7}\times2^{-\frac{2}{7}}\)
\(f^\prime(2)=\frac{12}{5\times2^{\frac{1}{5}}}-\frac{20}{7\times2^{\frac{2}{7}}}\approx\frac{12}{5\times1.1487}-\frac{20}{7\times1.2190}\)
\(f^\prime(2)\approx\frac{12}{5.7435}-\frac{20}{8.533}\approx2.09 - 2.34=- 0.25\)

Step3: Evaluate \(f^\prime(5)\)

Substitute \(x = 5\) into \(f^\prime(x)\).
\(f^\prime(5)=\frac{12}{5}\times5^{-\frac{1}{5}}-\frac{20}{7}\times5^{-\frac{2}{7}}\)
\(f^\prime(5)=\frac{12}{5\times5^{\frac{1}{5}}}-\frac{20}{7\times5^{\frac{2}{7}}}\approx\frac{12}{5\times1.3797}-\frac{20}{7\times1.5749}\)
\(f^\prime(5)\approx\frac{12}{6.8985}-\frac{20}{11.0243}\approx1.74 - 1.81=-0.07\)

Answer:

\(f^\prime(2)\approx - 0.25\)
\(f^\prime(5)\approx - 0.07\)