Question

which of the following is an equation of the line tangent to the graph of h(x)=x^4 - 2x^2+2x at the point where x = 1? choose 1 answer: a y=x + 1 b y=2x + 1 c y=2x - 1 d y=x - 1

Explanation:

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

Differentiate \(h(x)=x^{4}-2x^{2}+2x\) using the power - rule \((x^n)^\prime = nx^{n - 1}\).
\(h^\prime(x)=4x^{3}-4x + 2\)

Step2: Evaluate the derivative at \(x = 1\)

Substitute \(x = 1\) into \(h^\prime(x)\).
\(h^\prime(1)=4(1)^{3}-4(1)+2=4 - 4+2=2\)

Step3: Find the value of \(h(x)\) at \(x = 1\)

Substitute \(x = 1\) into \(h(x)\).
\(h(1)=1^{4}-2(1)^{2}+2(1)=1 - 2 + 2=1\)

Step4: Use the point - slope form of a line

The point - slope form of a line is \(y - y_1=m(x - x_1)\), where \((x_1,y_1)=(1,1)\) and \(m = 2\).
\(y - 1=2(x - 1)\)
\(y-1 = 2x-2\)
\(y=2x - 1\)

Answer:

C. \(y = 2x-1\)