Let f be a differentiable function everywhere and let p be a constant. Define a function g by g(x) = px − f(x). Geometrically, g measures the vertical distances between the graph of f and the straight line, y = px. We want to understand when g has extrema, that is, the largest or smallest vertical distances between the graph of the function f and a straight line. Prove that if g has an extremum value at x = c then ƒ must have a tangent line parallel to the line y = px.
Let f be a differentiable function everywhere and let p be a constant. Define a function g by g(x) = px − f(x). Geometrically, g measures the vertical distances between the graph of f and the straight line, y = px. We want to understand when g has extrema, that is, the largest or smallest vertical distances between the graph of the function f and a straight line. Prove that if g has an extremum value at x = c then ƒ must have a tangent line parallel to the line y = px.
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.5: Graphs Of Functions
Problem 36E
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![1. Let ƒ be a differentiable function everywhere and let p be a constant.
Define a function g by g(x) = px - f(x). Geometrically, g measures the vertical distances
between the graph of f and the straight line, y = px. We want to understand when g
has extrema, that is, the largest or smallest vertical distances between the graph of the
function f and a straight line.
Prove that if g has an extremum value at x = c then f must have a tangent line parallel
to the line y = px.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F64e22453-08e9-4901-b6d4-29904b54e3b3%2Ff073d393-5d81-4ad0-8862-a4ae3d6dc6e5%2Fz5smket_processed.png&w=3840&q=75)
Transcribed Image Text:1. Let ƒ be a differentiable function everywhere and let p be a constant.
Define a function g by g(x) = px - f(x). Geometrically, g measures the vertical distances
between the graph of f and the straight line, y = px. We want to understand when g
has extrema, that is, the largest or smallest vertical distances between the graph of the
function f and a straight line.
Prove that if g has an extremum value at x = c then f must have a tangent line parallel
to the line y = px.
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