Question

which of the following is an equation of the line tangent to the graph of f(x)=x^4 + 2x^2 at the point where x = 1? choose 1 answer: a y = 8x+2 b y = 8x - 5 c y = 3x+5 d y = 3x

Explanation:

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

The derivative of \(f(x)=x^{4}+2x^{2}\) using the power - rule \((x^n)^\prime = nx^{n - 1}\) is \(f^\prime(x)=4x^{3}+4x\).

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

Substitute \(x = 1\) into \(f^\prime(x)\): \(f^\prime(1)=4(1)^{3}+4(1)=4 + 4=8\). This is the slope of the tangent line.

Step3: Find the \(y\) - value of the function at \(x = 1\)

Substitute \(x = 1\) into \(f(x)\): \(f(1)=1^{4}+2(1)^{2}=1 + 2=3\). So the point of tangency is \((1,3)\).

Step4: Use the point - slope form \(y - y_1=m(x - x_1)\)

Here \(m = 8\), \(x_1 = 1\) and \(y_1 = 3\). So \(y-3=8(x - 1)\), which simplifies to \(y=8x-8 + 3=8x-5\).

Answer:

B. \(y = 8x-5\)