find the slope and the equation of the tangent line to the graph of the function at the given value of x…

Explanation:

Step1: Find the derivative of the function

Using the power - rule $(x^n)'=nx^{n - 1}$, if $y=x^{4}-17x^{2}+16$, then $y' = 4x^{3}-34x$.

Step2: Calculate the slope of the tangent line

Substitute $x = 1$ into $y'$. So, $m=y'(1)=4(1)^{3}-34(1)=4 - 34=-30$.

Step3: Find the y - coordinate of the point on the curve

Substitute $x = 1$ into $y=x^{4}-17x^{2}+16$. Then $y=(1)^{4}-17(1)^{2}+16=1 - 17 + 16=0$.

Step4: Find the equation of the tangent line

Use the point - slope form $y - y_{1}=m(x - x_{1})$, where $(x_{1},y_{1})=(1,0)$ and $m=-30$. So $y-0=-30(x - 1)$, which simplifies to $y=-30x + 30$.

Answer:

The slope of the tangent line is $-30$. The equation of the tangent line is $y=-30x + 30$.