4. f(x)= -6x + 6 f(x)= -3x² + 6x + c c = 3 th...
4. f(x)= -6x + 6 f(x)= -3x² + 6x + c c = 3 the graph of f, the derivative of f, is the line shown in the figure above. if f(0)=5, then f(1)= (a) 0 (b) 3 (c) 6 (d) 8 (e) 11 f(x)= -3x² + 6x + 3 f(1)= -3 + 6 + 3
Answer
# Explanation:
## Step1: Find the antiderivative of \(f^{\prime}(x)\)
Given \(f^{\prime}(x)=-6x + 6\), the antiderivative \(f(x)=\int(-6x + 6)dx=-3x^{2}+6x + C\)
## Step2: Determine the value of \(C\)
Since \(f(0) = 5\), substitute \(x = 0\) into \(f(x)=-3x^{2}+6x + C\). We get \(f(0)=-3(0)^{2}+6(0)+C=5\), so \(C = 5\)
## Step3: Calculate \(f(1)\)
Substitute \(x = 1\) into \(f(x)=-3x^{2}+6x + 5\). Then \(f(1)=-3(1)^{2}+6(1)+5=-3 + 6+5=8\)
# Answer:
D. 8