9.5: For ​f(x)=2x4−4x2+5 find the following. ​ (A) f′(x) ​(B) The slope of the graph of f at x=2 ​(C) The equation of the tangent line at x=2 ​(D) The​ value(s) of x where the tangent line is...

Question

For ​f(x)=2x⁴−4x²+5 find the following.

​

(A) f′(x)

​(B) The slope of the graph of f at x=2

​(C) The equation of the tangent line at x=2

​(D) The​ value(s) of x where the tangent line is horizontal

Answer 1

(A) f'(x) = 8x³ - 8x

(B) the slope of the graph at x = 2 is equel f'(2) = 8·2³ - 8·2 = 48

(C) Tangent point f(2) = 2·2⁴ - 4·2² + 5 = 21; so, (2, 21) is tangent point.

Equation of tangent line: y = 21 + 48(x - 2); y = 48x - 75

(D) Tangent line horizontal where f'(x) = 8x³ 8x = 0; x(x - 1)(x + 1) = 0; x = 0, x = 1 and x = - 1