find the antiderivative for each function when c equals 0. check your answers by differentiation. (a)…

Explanation:

Step1: Recall power - rule for antiderivatives

The antiderivative of $x^n$ is $\frac{x^{n + 1}}{n+1}+C$ for $n\neq - 1$.

Step2: Find antiderivative of $h(x)=\frac{3}{4}x^{-\frac{1}{4}}$

Let $n =-\frac{1}{4}$. Then $H(x)=\frac{3}{4}\times\frac{x^{-\frac{1}{4}+1}}{-\frac{1}{4}+1}$. Simplify the exponent and the fraction: $-\frac{1}{4}+1=\frac{3}{4}$, and $\frac{3}{4}\times\frac{x^{\frac{3}{4}}}{\frac{3}{4}}=x^{\frac{3}{4}}$.

Step3: Find antiderivative of $g(x)=\frac{1}{4}x^{-\frac{3}{4}}$

Let $n =-\frac{3}{4}$. Then $G(x)=\frac{1}{4}\times\frac{x^{-\frac{3}{4}+1}}{-\frac{3}{4}+1}$. Simplify: $-\frac{3}{4}+1=\frac{1}{4}$, and $\frac{1}{4}\times\frac{x^{\frac{1}{4}}}{\frac{1}{4}}=x^{\frac{1}{4}}$.

Step4: Find antiderivative of $k(x)=-\frac{1}{4}x^{-\frac{5}{4}}$

Let $n =-\frac{5}{4}$. Then $K(x)=-\frac{1}{4}\times\frac{x^{-\frac{5}{4}+1}}{-\frac{5}{4}+1}$. Simplify: $-\frac{5}{4}+1=-\frac{1}{4}$, and $-\frac{1}{4}\times\frac{x^{-\frac{1}{4}}}{-\frac{1}{4}}=x^{-\frac{1}{4}}$.

Answer:

(a) $H(x)=x^{\frac{3}{4}}$ (b) $G(x)=x^{\frac{1}{4}}$ (c) $K(x)=x^{-\frac{1}{4}}$