For What Values Of $c$ Does the Curve Have Maximum and Minimum Points for the and Minimum

question 60

Multiple Choice

For what values of $c$ does the curve have maximum and minimum points for the given function $f ( x ) = c x ^ { 4 } - 5 x ^ { 2 } + 1 ?$
Select the correct answer.

• A) For \(c = 0 , f ( x ) = - 5 x ^ { 2 } + 1\), a parabola whose vertex \(( 0,2 )\), is the absolute maximum.
• For \(c > 0 , f ( x ) = c x ^ { 4 } - 5 x ^ { 2 } + 1\), opens downward with one minimum points.
• For \(c < 0\), the graph opens upward, and has an absolute maximum at \(x = 0\) and no local minimum.
• B) For \(c = 0 , f ( x ) = 5 x ^ { 2 } + 1\), a parabola whose vertex \(( 0,2 )\), is the absolute maximum.
• For \(c > 0 , f ( x ) = c x ^ { 4 } + 5 x ^ { 2 } + 1\), opens upward with two minimum points.
• For \(c < 0\), the graph opens downward, and has an absolute minimum at \(x = 0\) and no local minimum.
• C) For \(c = 0 , f ( x ) = 5 x ^ { 2 } + 1\), a parabola whose vertex \(( 0,3 )\), is the absolute maximum.
• For \(c > 0 , f ( x ) = c x ^ { 4 } - 5 x ^ { 2 } + 1\), opens upward with two minimum points.
• For \(c < 0\), the graph opens downward, and has an absolute minimum at \(x = 0\) and no local minimum.
• D) For \(c = 0 , f ( x ) = - 5 x ^ { 2 } + 1\), a parabola whose vertex \(( 0,1 )\), is the absolute maximum.
• For \(c > 0 , f ( x ) = c x ^ { 4 } - 5 x ^ { 2 } + 1\), opens upward with two minimum points.
• For \(c < 0\), the graph opens downward, and has an absolute maximum at \(x = 0\) and no local minimum.
• E) For \(c = 0 , f ( x ) = - 5 x ^ { 2 } + 1\), a parabola whose vertex \(( 0 , - 1 )\), is the absolute maximum.
• For \(c > 0 , f ( x ) = c x ^ { 4 } + 5 x ^ { 2 } - 1\), opens downward with two maximum points.
• For \(c < 0\), the graph opens upward, and has an absolute minimum at \(x = 0\).

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Definitions:

Projective Test

A psychological assessment tool that uses ambiguous stimuli to elicit responses that reveal aspects of an individual’s personality.

Thematic Apperception Test

A projective psychological test where individuals are presented with ambiguous pictures and asked to create stories, revealing their underlying motives, feelings, and attitudes.

Validity

Refers to the degree to which a test or instrument measures what it claims to measure.

MMPI

The Minnesota Multiphasic Personality Inventory, a comprehensive psychological assessment tool used to diagnose mental disorders and assess personality structure.