how to find the coordinates of the tangent at 45 degrees of a polynomial function

Question

I'm stuck! I need to find the coordinates of the tangent at $45$ degrees of this function: $y=0.3582x^5-2.2501x^4+4.6235x^3-4.5377x^2+2.4503x+.3513$

I need to know where is the inflection point at $45$ degrees on the graph. By looking at the graph in Excel ($x$ Axis varies from $0%$ to $100%$ only), the $y$ ($y$ Axis varies from $0%$ to $100%$ too), coordinates of the tangent at $45$ degrees should be between 75% and 80%.

I have found the derivative: $y=1.791x^4-9.000x^3+13.8705x^2-9.0754x+2.4503$

I have been told that the slope of a tangent at $45$ degrees is $1$, therefore, if I replace "$x$"$=1$ in the derivative, I will get the "$y$" value.

Now what should I do and is the process accurate?

Tks,

Answer 1

Since the setting of the problem is numerical, i will give also a rather numerical answer. We have to solve the equation $$ f'(x)=1\ . $$ The following lines of sage code

R.<x> = PolynomialRing(RR)
f = 0.3582*x^5 - 2.2501*x^4 + 4.6235*x^3 - 4.5377*x^2 + 2.4503*x + 0.3513
A = ( diff(f,x) - 1 ).roots(multiplicities=False)
print A
plot( [f] + [f(a)+(x-a) for a in A ], (x, -0.8, 3.6), aspect_ratio = 1 )

are finding the following approximative roots of $f'(x)-1$, in code diff(f,x) - 1:

[0.228077877625767, 2.95170314961792]

The plot gives the picture: